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Full text of "Quanta - A Handbook of Concepts"


CiniSIlT SUES 

quanta 

of eancepts 
R W. Atkins 



Oxford Chemistry Series 



General Editors 

P. W. ATKINS J. S. E. HOLKER A. K. HOLLIDAY 



Oxford Chemistry Series 



1972 

1. K. A. McLauchlan: Magnetic resonance 

2. J. Robbins: Ions in solution (2): an introduction to electrochemistry 

3. R. J. Puddephatt: The periodic table of the elements 

4. R. A. Jackson: Mechanism: an introduction to the study of organic reactions 

1973 

5. D. Whittaker: Stereochemistry and mechanism 

6. G. Hughes: Radiation chemistry 

7. G. Pass : Ions in solution (3) : inorganic properties 

8. E. B. Smith: Basic chemical thermodynamics 

9. C. A. Coulson : The shape and structure of molecules 

10. J. Wormald: Diffraction methods 

11. J. Shorter: Correlation analysis in organic chemistry: an introduction to linear free-energy relationships 

12. E. S. Stern (ed): The chemist in industry (I): fine chemicals for polymers 

13. A. Earnshaw and T. J. Harrington: The chemistry of the transition elements 

1974 

14. W. J. Albery : Electrode kinetics 

16. W. S. Fyfe: Geochemistry 

17. E. S. Stern (ed): The chemist in industry (2): human health and plant protection 

18. G. C. Bond: Heterogeneous catalysis: principles and applications 

19. R. P. H. Gasser and W. G. Richards: Entropy and energy levels 

20. D. J. Spedding: Air pollution 

21. P. W. Atkins: Quanta: a handbook of concepts 

22. M. J. Pilling: Reaction kinetics 



P. W. ATKINS 

FELLOW OF LINCOLN COLLEGE. OXFORD 



Quanta 

a handbook of concepts 



Clarendon Press ■ Oxford ■ 1974 



Oxford University Press, Ely House, London W. 1 

GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON 
CAPE TOWN IBADAN NAIROBI DAR ES SALAAM LUSAKA ADDIS ABABA 
DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA 
KUALA LUMPUR SINGAPORE HONG KONG TOKYO 



ASEBOUND ISBN 19 855493 1 
PAPERBACK ISBN 19 855494 X 



TEXT © OXFORD UNIVERSITY PRESS 1974 
ILLUSTRATIONS © P. W. ATKINS 1974 



All rights reserved. No part of this publication may be reproduced, 
stored in a retrieval system, or transmitted, in any form or by any means, 
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the prior permission of Oxford University Press 



ACCESSION no. 



l^- 











PRINTED IN GREAT BRITAIN 

BY FLETCHER & SON LTD., NORWICH 



Preface 



Here is a book that attempts to explain the quantum theory 
without mathematics. 

Of course, I agree that quantum theory has an inescapable 
mathematical structure; I agree that the full precision of the 
theory, and its richness, can be conveyed only in the language 
of mathematics; I also agree that to make substantial con- 
tributions to quantum theory one needs a firm grasp of its 
mathematical basis. Nevertheless, I also believe that math- 
ematical precision is not what everyone is after— not everyone 
has that kind of interest, nor indeed that amount of time. 

Most of us have our pictures of the concepts of quantum 
theory-we have some way of visualizing orbitals, transitions, 
etc.-and those of us who do research are more often than not 
guided in broad outline by some visualizable model of the 
system we are trying to describe. This book contains my ways 
of thinking about the concepts of quantum theory. The 
pictures, like all analogies and models, are only a partial 
representation of the true situation; but I hope they contain 
the heart of the matter, and enable the reader to understand 
each idea at a physical, rather than mathematical, level. 

I hope that a broad selection of people will find the book 
useful. I have had in mind both the student of chemistry who 
at all levels of his studies encounters unfamiliar, little- 
understood, or half-forgotten concepts, and his teacher who 
is pressed for an explanation. I hope that by reference to this 
book all will be provided with just enough information to 
make the concept clear and perhaps even to be stimulated to 
find our more. If my explanation is inadequate, my bibli- 
ography will direct the dissatisfied along a trail of others' 
explanations. 

Each entry is intended to explain, in plain language, the 
physical content of its topic. Most are illustrated by examples, 
and, where appropriate, counter examples. Where the entry 
draws on information contained elsewhere in the book, or 
where further development of a topic is desirable, or where the 
reader might be felt to require more background, I have 
labelled a word with an unobtrusive * to signify that it is in 
the book. For a good reason I cannot now remember, the 



label precedes the indicated word; the labelled word does not 
always correspond to an exact entry, but its sense should be 
sufficient to indicate the appropriate entry without ambi- 
guity. Unlike most dictionaries this has an index: some 
concepts are buried inside others. 

Where I feel it desirable that some mathematics be intro- 
duced (for example when it seems helpful to have a collection 
of formulae to hand, e.g. in "perturbation theory), I have used 
a system of Boxes and Tables. A Box is part of the text, and 
contains handy formulae: these are confined to Boxes in order 
that they be present but held away from the description in the 
text. A Table contains more detailed information (sometimes 
of mathematical expressions not required for a reading of the 
text) which I judge it helpful to have in a volume of this kind 
(so that it can be used at more than just its principal level of 
qualitative description), and which I judge would have intruded 
too vigorously into the main text. These Tables, which are 
collected at the end of the volume, also contain a selection of 
experimental data. I think it important that one has an idea of 
the size of physical quantities, and their trends, and so these 
numbers are included to sketch out the range of experimental 
data. The Tables are by no means exhaustive: they simply peg 
out the terrain. 

Nearly every entry is followed by Questions. These are set 
with a double purpose and are graded from the trivial to the 
slightly tough. The early parts of each are intended to focus 
the reader's mind on key points made in the preceding entry. 
Most of these simple questions can be answered by referring to 
the entry. The second purpose is to bring a tiny amount of 
mathematics into the book in an unobtrusive way: the harder 
parts of the questions (which when they occur are often 
labelled 2) invite the reader to make his own mathematical 
exposition of the entry, and contain hints and guidance to that 
end. The answers to the questions are mostly in the text, in a 
Box, or on a Table; if that is not so. Further information points 
the way. 

Each entry contains a section headed Further information. 
This is a guide to the literature, and in it will be found refer- 



ences to books and articles where the reader can turn for more 
information, the development of the topic, and the absent 
mathematics. This section is not exhaustive (it contains works 
I have found helpful) but once a trail is indicated the literature 
is more easily penetrated. In this section I have attempted to 
list the books in order of complexity so that the reader can 
make a progression through difficulty. There are however, 
two important exceptions to this organization. The first is the 
reference to MQM. This is my Molecular quantum mechanics 
(Clarendon Press, 1970); many of the topics are treated in 
more detail in that book, and a reference to the appropriate 
section is included at the head of each Further information. 
The second exception reflects the fact that this book is a part 
of the Oxford Chemistry Ser/'es-although its size makes it a 
sport— and I have made a point of directing the reader to the 
other books in the series where the topics are developed: these 
are denoted OCSn and listed together at the start of the bibli- 
ography. OPSn books are the first few of the analogous 
Oxford Physics Series, and are listed likewise. 

In a few places I have aimed at a higher level of exposition 
than the rest, especially when I have been unable to give what 
I consider to be a satisfactory explanation at the low, quali- 
tative level at which in general I have aimed. Let me stress, 
however, that I do not use the word 'low' pejoratively: it is, I 
believe, as important for a chemist to have a physical intuition 
about the behaviour of submicroscopic phenomena as it is for 
him to be able to manipulate the mathematics of the descrip- 
tion. In my 'low' level descriptions I am attempting to train 
this intuition. I shall not draw attention to the entries I con- 
sider to be at this higher level: if they are not noticed, so much 
the better. If they are noticed, and found bewildering, the ° on 
keywords will guide the reader to quieter waters. In this con- 
nexion, however, I must draw attention to one distressing 
feature of Nature. As in a conventional dictionary, where the 
unlikely concepts of aardvark, aasvogel, and ablet bring con- 
fusion and difficulty in unnatural proportion into A, so by the 



same quirk does Nature concentrate difficulty into the A of 
Quanta. I find the entries of A more difficult than the entries 
of B; perhaps you will too. (My message is that reading A— as a 
browser might feel inclined-gives, I think, a false impression 
of the overall level of the entries.) 

Two other points are worth making. The first is an excuse. 
In an attempt to keep the price of the book low (at least by 
the standards we are being trained to expect) all the diagrams 
are my own sketches. All were done on a balcony in Italy one 
summer, and each jolt, smudge, or splash tells its story of 
inquisitive mosquito, local ant, or homing wasp. Please forgive 
their generally amateur appearance. The second point concerns 
those to whom notes (such as these entries) are anathema, 
those who want material for more books, or those who 
seek an essay. Each should notice that the approximately 
200 entries can be permuted in some 10 300 different ways: 
surely out of these some decent books can be wrought? 

The real purpose of writing a preface is to come to the part 
that gives most pleasure: the thanking of all those who have 
contributed to the production of the book and easing its grim 
labour. A quite outstanding contribution-beyond the call of 
duty and reasonable expectation-has been made by the 
Clarendon Press and its anonymous officers. Their assistance 
ranged from advice and help from the drawing office, 
through detailed and lengthy discussion of presentation, 
to careful and ingenious production. Two of my research 
students deserve my thanks: Michael Clugston bent his per- 
ceptive eye on the proofs and saved me from much shame, and 
John Roberts spent time coaxing contours out of computers; 
to both am I most grateful. To the others I express my thanks 
for accepting neglect without overt complaint. The typing of 
the whole obscure manuscript (as my own typed original is 
better regarded) was done briskly and efficiently by 
Mrs. E. Price and Mrs. M. Long, both of whom deserve at 
least what immortality this page can provide. 

P.W.A. 



A 



ab initio. Not the whole of quantum chemistry is conducted 
in Latin; the small portion to which it has by convention been 
confined, as an ironic meeting of cultures, belongs to those 
whose business is computers. Ab initio, roughly translated, 
means from scratch, and is applied to the molecular-structure 
computations that abhor the inclusion of empirical data and 
attempt to calculate from first principles, which for our pur- 
pose are the "Schrodinger equation and the method of •self- 
consistent fields. 

Further information. Richards and Horsley (1970} have 
prepared a short, simple guide to ab initio calculations, and 
work through a number of examples. They also discuss the 
relation of such calculations to the semi-empirical methods in 
which approximations and empirical data are introduceo into 
"self-consistent field calculations. 



adiabatic process. The term adiabatic is used in both thermo- 
dynamics and quantum mechanics, and the uses are analogous. 
In the former it signifies that the process is occurring without 
exchange of heat with the environment {as this implies that 
there is no change of entropy, the process is often called 
isentropic). In the latter it signifies that a change is occurring 
so that the system makes no transition to other states. 

Consider, for example, a hypothetical 'hydrogen atom 
with a variable nuclear charge. If the atom is initially in its 
ground state, and the nuclear charge is increased extremely 
slowly, the electron will be sucked in closer to the nucleus; 
but the atom remains in its ground state, and by the time 



that Z= 2 the system is a ground state He + ion: this is there- 
fore an adiabatic process. Conversely, if the nuclear charge of 
an atom is changed suddenly (for example, by the emission of 
an electron in /3-decay), the bound electron finds itself in a 
different nuclear potential but with its original spatial distri- 
bution: this distribution can be expressed as a mixture of 
°wavef unctions of the new atom, and so in this impulsive or 
nonadiabatic transition the system is knocked into a range of 
states of the final system. The slow compression of a "particle 
confined in a square well is another example of an adiabatic 
transition, for if the system is in the nth level of the original 
box it will be in the nth level of the new, smaller box if the 
compression is infinitely slow. 

Further information. See MQM Chapter 7. An account of 
adiabatic transitions in terms of "perturbation theory will be 
found in §76 of Davydov (1965), who derives the condition 
that a motion is adiabatic if the perturbation V(t) changes so 
slowly that d V/dt is much smaller that the energy separations 
in the vicinity of the initially occupied state: jdV/dr| <§ 
(A£ } 2 /h. See also §50 of Bohm (1951 ) for a pleasing discussion 
with straightforward mathematics. Recent review articles on 
adiabatic and nonadiabatic processes in molecules have been 
published by Ko/os (1970) and Nikitin (1970) respectively. 
See "non-crossing rule. Thermodynamic adiabaticity is dis- 
cussed by Smith in his Basic chemical thermodynamics 
(OCS 8) and entropy is related to the distribution of particles 
among energy levels in Gasser and Richards' Entropy and 
energy levels (OCS 19). This is the key connexion between 
the thermodynamic and quantum-mechanical uses of the term 



2 alternant hydrocarbon 

adiabatic: an unchanged distribution of particles among states 
is an isentropic situation. 

alternant hydrocarbon. A chain of carbon atoms in a 
hydrocarbon that can be labelled alternately by a Star, no star, 
a star, and so on, such that no two stars are neighbours nor 
two unstarred atoms are neighbours when the labelling is 
complete, is an alternant hydrocarbon. An example is propene, 
which could be labelled C*~C-C*, or C-C*~C; another 
example is benzene (1), and an example of a non-alternant 
hydrocarbon is azulene 





(2). Alternants are distinguished by 
several electronic characteristics: 

(a) To every bonding orbital of energy — E there is a comp- 
lementary 'antibonding orbital of energy +E: the bonding 
and antibonding orbitals are arranged symmetrically about 
zero energy (Fig. A1). 

(b)ln an even-alternant the number N of carbon atoms is 
even, and usually the numbers* of starred atoms is equal to 




FIG. Al. Energy levels ol 3 typical alternant (benzene) and a typical 
non-alternant (cyclopentadienyl). Note the symmetric disposition of the 
levels in the former. 



the number n of unstarred. In this case there are ^N bonding 
orbitals and ^/v antibonding orbitals. If n* and n are unequal 
there are In*— nl nonbonding orbitals inserted between a 
symmetrical array of n bonding and n antibonding orbitals 
(forn <n*). In an odd-alternant the number of carbon atoms 
is odd, and usually there is one more starred atom than un- 
starred (n* — d+1 1, and one nonbonding orbital is inserted 
between the symmetrical array of n bonding and n antibonding 
orbitals {n+n * ^/V). 

(c)The distribution of electrons is more uniform in altern- 
ant hydrocarbons than in non-alternants. This property is 
expressed quantitatively by the Coutson-Rushbrooke theorem 
which states that the 7r-electron "charge density on every atom 
in the ground state of an alternant hydrocarbon is unity {each 
carbon has just one TT-electron associated with it). 

(d) In odd-alternants the electron density of the correspond- 
ing cation or anion may be deduced very simply by considering 
the form of the nonbonding orbital, for it is this orbital from 
which an electron is taken to form the cation, or to which one 
is added to give the anion, and in the neutral hydrocarbon the 
charge, distribution is uniform. The form of the nonbonding 
orbital may also be deduced, virtually by inspection, by rely- 
ing on the following device. Star the atoms in such a way as to 
get the maximum number of non-neighbour stars, then the 
amplitude of the orbital on each unstarred atom is zero. 
Furthermore, the sum of the coefficients of orbitals on starred 
atoms attached to a given unstarred atom is zero. This gives 
the relative size of all the coefficients; to get their absolute 
size the orbital is "normalized. The charge density on each 
atom is obtained by squaring the coefficients. 

The stability of even-alternants (such as 'benzene) can be 
understood in terms of the preceding properties. In particular, 
in an /V-atom even-alternant each atom provides one TT-electron; 
each of the ^/V bonding orbitals may accommodate two 
electrons ( -Paul! principle), and so only the bonding orbitals 
are occupied. This structural stability is further protected from 
reactive attack by the uniformity of the charge distribution, 
which provides no centres of attraction for potential reagents. 

Questions. How is an alternant hydrocarbon distinguished 
from a non-alternant? Which of the following hydrocarbons 
are alternant: ethylene (ethene), butadiene (buta-1, 3-diene), 



angular momentum 



cyclobutadiene, benzene, naphthalene, anthracene, azulene, 
cycfo-octatetraene, phenylmethyl (the benzyl radical), 
cyclopentadienyl? What properties can you predict for the 
alternant hydrocarbons of this list? State the Coulson- 
Rushbrooke theorem. Use the form of the "benzene molecular 
orbitals {p. 20) to confirm that the charge density in benzene 
is uniform and in accord with the theorem. Evaluate the form 
of the nonbonding molecular orbital in the phenylmethyl 
radicai by the method described in note (d). (You should find 
the coefficients 2A/"7 on CH 2 , and a collection of ±1//7 and 
elsewhere.) Deduce from this the charge distribution in the 
cation 0CH2 and the anion 0CH2. Can you deduce anything 
about the chemical reactivity of phenylmethyl? 

Further information. See Coulson's The shape and structure of 
molecules {OCS 9), A helpful account of alternant molecules 
will be found in Chapter 9 of Roberts (19616), in §2-6 of 
Streitweiser (1961 ), in Pilar (1968), and in Salem (1966). A 
book devoted to them is that of Pauncz (1967). The spectro- 
scopic properties are described by Murrell (1971 ) and reviewed 
by Hall and Amos (1969), who furnish further directions to 
the literature. The calculation of the electronic structure of 
alternants has been described by Parr (1963), whose book 
includes reprints of some of the original papers, by Dewar 
(1969), Popie and Beveridge (1970), and Murrell and Harget 
(1972). Tables of molecular-orbital coefficients and energies 
have been prepared by Coulson and Streitweiser (1965). For 
a proof of the Coulson-Rushbrooke theorem see Coulson and 
Rushbrooke (1940) and a review article by Coulson (1970). 

angular momentum. The angular momentum of an object 
in classical mechanics is /to, where / is its moment of inertia 
and oj its angular velocity (in radians per second): a big object 
(with a big moment of inertia) need rotate only slowly (have 
small angular velocity) in order to achieve the same angular 
momentum as a small object rotating rapidly. In classical 
mechanics an object may rotate with any angular momentum; 
but in quantum theory the magnitude of the angular 
momentum of any body is "quantized and limited to the 
values {/(/+1)] !4 ri, where/ is a non-negative (zero or positive) 
integer or half-Integer (0, j, 1, . . . ). Only one component of 
this angular momentum may be specified (that is, we may 



state the angular momentum of a body about only one axis), 
and its values are limited to mh, where m =/,/— 1, . . . — j. 
This implies that, contrary to the classical situation, a rotating 
body may take up only a discrete sequence of orientations 
with respect to any one selected axis: the quantization of 
orientation is called space quantization. 

A convenient representation of the angular momentum is as 
a vector of length [/(/+1)] w , see Fig. A2, which may take up a 
discrete series of orientations as depicted in Fig. A3 for a body 
with/=2; this is the basis of the -vector model of the atom. 




FIG. A2. The eiassical angular momentum and its representation by a 
vector of specified projection on thez-axis. 

Since only one component of this vector can be specified (and 
conventionally this is taken to be the^-component) the 
azimuth of the vector (its orientation in the xy-plane) is inde- 
terminate; the cone of possible orientations represents the 
property of -precession. 

The value of the quantum number for -orbital angular 
momenta (the momenta arising from the spatial distribution 
of the particle) is confined to integers; it is convention to use 
the letter 8 to denote the orbital angular momentum quantum 
number, and so perforce £ is confined to non-negative integral 
values. The intrinsic angular momentum of a particle, its 'spin, 



angular momentum 
1 



length v^jC jH>' 




classical trajectory 



FIG. A3. An angular momentum with/' 2 can take only five {2/ + 1 1 
orientations in space according to quantum mechanics, but all orien- 
tations according to classical mechanics. The discrete orientations are 
illustrated. 



is described by a quantum number that may. have either 
integral or half-integral values, and which is normally denoted 
s (or / for nuclei). 

If a system contains two sources of angular momentum its 
total angular momentum is also quantized and restricted to 
magnitudes [/(/+1 )] h, with/ confined to the values/[ 4/j , 
/] +-/j— 1, . . . l/j— fjl, where/] and/ 2 are the quantum 
numbers of the component momenta. This sequence of 
numbers is known as the Clebsch-Gordon series. As an 
example, an electron with spin s — j and in an orbital with 
£ = 1 constitutes a system with two sources of angular 
momentum. The total angular momentum of the electron 

t 1 3 1 

may take on the values given by/= 1+— 1+j— 1, or ^ and ^, 
depending on the relative orientation of the two momenta: 
if the individual momenta are parallel the total momentum is 
high (/— j), if they are opposed it is low (/—j). When the 
system contains several sources of momentum the overall 
angular momentum is quantized and constructed by coupling 
/i and/ 2 , then/ 3 to their resultants, and so on, each step 
being in accord with the Clebsch-Gordon series. 

In quantum mechanics an angular momentum can be de- 
fined in terms of a set of -commutation rules of the appro- 
priate 'operators; any set of operators that satisfies the 
commutation rule [/ ,/ ] = in/ is called an angular momen- 
tum, and the properties outlined above are common to all 



such creatures. In this way the theory of angular momentum 
expands to embrace intrinsic properties of systems, such as 
their -spin and their charge. 

Questions. What are the features of quantized angular momen- 
tum? What magnitudes of angular momentum correspond to 
quantum numbers equal to ^, 1,2, 10 M ? Draw a vector rep- 
resentation of an angular momentum and take £ = 1, m = 1, 
0, —1; also draw the classical motion of a particle correspond- 
ing to the three values of m. Consider the vector representation 
of the angular momentum of a bicycle wheel: is it possible to 
ride a bicycle strictly perpendicular to the road? At what 
velocities would this quantum wobble be intolerable? The spin 
of a "photon is 1: what is the magnitude of its intrinsic angular 
momentum? What angular momenta can arise from coupling 
the spin of an electron with its -orbital momentum in a 
d-orbital (£ = 2)? What states of total orbital momentum can 
be obtained by coupling the momenta of two p-electrons, two 
d-electrons, a p- and a d-electron, three p-electrons? 

Further information. See MQM Chapter 6 for a detailed dis- 
cussion of angular momentum. An interesting account which 
emphasizes the connexion between classical and quantal 
angular momenta is given by Kauzmann {1957). Books dealing 
specifically with the quantum theory of angular momentum, 
and ranging from the moderately accessible to the very 
difficult, include those by Brink and Satchler {1968), Rose 
{1957), Edmonds (1957), Judd (1963), and Beidenharn and 
van Dam (1965); the last contains a number of important 
original papers. The relation of angular momentum to the way 
a system changes as it is rotated is described in MQM Chapter 
6, in Tinkham (1964), which is a good introduction to the 
connexion between the symmetry of a system and its angular 
momentum, and in angular momentum books. The wavefunc- 
tion for a state of coupled angular momentum may be ex- 
pressed as a combination of the wavefunctions of the 
contributing uncoupled states; the coefficients of the combi- 
nation are the "Wigner coefficients. The formal relation of 
angular momentum to other properties of a system, such as 
its charge, is described qualitatively by Lipkin (1965) and in 
more, but not excessive, detail by Lichtenberg (1970). 



anharmonicity 



anharmonicity. There are two types of anharmonicity: 
mechanical a n d e I ect ri ca I , Mechanics I anharmonicity (com- 
monly referred to simply as anharmonicity) occurs when an 
oscillator is in a potential that is not purely parabolic, so that 
the restoring force is not strictly proportional to the displace- 
ment. The energy levels in such a case are no longer strictly 
those of a "harmonic oscillator, and if the nature of the anhar- 
monicity is to lower the potential at large displacements the 
levels converge at high quantum numbers, as shown in Fig. A4. 
The lines in a "vibrational spectrum in the presence of anhar- 
monicity are therefore no longer evenly spaced. Another effect 
of mechanical anharmonicity is to ruin the "selection rules for 
a harmonic oscillator: if the molecular vibration is anharmontc 
it is unreasonable to expect rules developed for a harmonic 
oscillator to be applicable. Therefore some forbidden tran- 
sitions become allowed, and harmonics of the fundamental 
transitions are observed (corresponding to changes in the 
oscillator quantum number by +2, +3, etc.). These tran- 
sitions increase in intensity with the extent of anharmonicity 
in the potential. 

The intensity of transitions, and the failure of the harmonic 
oscillator selection rules, are also affected by the electrical 
anharmonicity, which is the name applied when the dipole 
moment of the molecule depends non- linearly on the displace- 
ment. The selection rules are normally calculated on the basis 
of the assumption that as the molecule is stretched the dipole 
moment changes linearly with the displacement (that is, the 
change in dipole moment is directly proportional to the dis- 
placement): if this is not so it is possible for the non-linear 





term in its true dependence, and in particular the term 
quadratic in the displacement, to induce transitions by 
Av = ±2, Thus electrical anharmonicity can cause intensity 
changes in the vibrational spectrum of a molecule similar to 
those caused by the mechanical anharmonicity; but in con- 
trast to the latter it does not affect the energy levels them- 
selves. 

A further effect of anharmonicity on the intensities in a 
•vibrational (infrared) spectrum of a molecule arises from its 
ability to mix together vibrations of various symmetries. In 
the harmonic approximation one encounters the "normal 
modes of vibration: these constitute a set of independent 
vibrational motions of the molecule. When anharmonicity is 
present the normal modes are no longer independent, and 
vibrational energy in one may leak into others. Interpreted 
quantum- mechanically, we say that the wavef unction for a 
normal mode mixes with, and therefore acquires some of the 
characteristics of , some of the other normal modes. An import- 
ant case in which normal modes mix as a result of anharmon- 
icity is Fermi resonance, which is a mechanism whereby the 
simultaneous excitation of two vibrational modes (which 
appears in the spectrum as a combination band) is permitted 
because nearby (in energy) there is a fundamental excitation 
frequency of another, allowed vibrational mode. The anhar- 
monicity in the molecular motion endows the mixture of 
vibrational modes with some of the characteristics of the 
allowed fundamental, and so the transition to the combin- 
ation becomes allowed. The extent to which it becomes 
allowed depends on the amount of anharmonicity and the 



FIG. A4. Anharmonic potentials 
(shown in colour) distort the even 
spacing of the levels in a harmonic 
potential (black). In (a) a broader 
potential reduces the separation 
by different amounts; in Ibl a 
narrow potential separates the 
levels further; and in (c) the 
complicated anharmonicity 
typical of a chemical bond is 
illustrated. 




6 antibonding 

closeness in frequency of the combination and fundamental 
energies, and is greatest when they are in "resonance. 

Yet another manifestation of anharmonicity is through its 
effect on the moment of inertia of a molecule, and through 
that on the molecule's "rotational motion and spectrum. A 
harmonically vibrating molecule has the same mean size what- 
ever its vibrational state; but an anharmonic molecule would 
tend to swell slightly, and to change its moment of inertia, as 
it is excited to higher vibrational states (see Fig. A4c). The 
dependence of the moment of inertia on the vibrational state 
affects the structure of the "branches in the vibration- rotation 
spectrum. 

Questions, 1 . What types of anharmonicity exist, and what do 
they affect? Discuss the effect of replacing a harmonic oscil- 
lator potential by one that is almost parabolic, but (a) gets 
broader, (b) gets narrower as the displacement increases. What 
is expected to be the form of the anharmonicity for a typical 
bond stretch? Discuss the vibrational potential for the out-of- 
plane vibration of a planar molecule, with special reference to 
the anharmonicity. What previously forbidden transitions 
become allowed in the presence of anharmonicity? What tran- 
sitions does electrical anharmonicity permit? What effect does 
electrical anharmonicity have on the energy of an oscillator? 
What effect does the presence of anharmonicity have on the 
symmetry selection rules? What is the group-theoretical inter- 
pretation of this? What is a combination band, and why does 
Fermi resonance endow it with intensity? What effect is there 
on the intensity of the allowed fundamental when it takes part 
in Fermi resonance? What happens to the latter's energy? What 
group -theoretical reason accounts for our stressing the role of 
a combination band in Fermi resonance rather than simply 
another fundamental? (Consider the symmetry of the anhar- 
monic part of the molecular energy .J 
2. Consider a potential of the form ^kx 2 + ax. Sketch the 
form of the potential on the assumption that a is small, and 
apply second-order "perturbation theory to the calculation of 
the effect of the linear anharmonicity on the potential. Show 
that the electric dipole moment may depend both linearly and 
quadratically on the displacement of the molecule from 
equilibrium (use a Taylor expansion) and that the quadratic 
term can induce transitions disallowed in a harmonic oscillator. 



Use the properties of the -harmonic oscillator in Table 1 1 on 
p. 273. 

Further information. See MQM Chapter 10 for a discussion of 
anharmonicity, and a discussion of the role of symmetry in 
governing what vibrations Fermi resonance may mix together. 
Woodward (1972) has a helpful qualitative discussion in §18-5 
and |22*10; so too do Brand and Speakman (1960) in §6*7 
and King (1964) in §5-5. See also Barrow (1962), Whiffen 
(1972), Gans (1971), and Wilson, Decius, and Cross (1955). 
Extreme anharmonicity leads to dissociation: the extrapolation 
of anharmonicity to this limit is discussed in Chapter 5 of 
Gaydon (1968). 

antibonding. An antibonding orbital is one that, when occu- 
pied, tends to induce dissociation. Imagine the Is-atomicorbitals 
on two "hydrogen atoms which are being brought together, 
and suppose that the signs of the amplitudes of the two wave- 
functions are opposite. The effect of bringing the atoms 
towards each other is to slide the region of positive amplitude 
of one wave into the region of negative amplitude of the other, 
and, just as in the case of conventional wave phenomena, the 
waves interfere destructively and the total wave amplitude in 
the region of overlap is diminished. The square of this ampli- 
tude determines the probability of finding the electrons in a 
particular region, and so the effect of bringing the orbital s 
together with opposite phases (signs) is to diminish the 
electron density in the internuclear region. This has an adverse 
effect on the energy of the molecule (because the internuclear 
region is the best place to put the electrons, for then their 
interaction with the two nuclei is the most favourable), and 
the molecule formed in this way will have an energy greater 
than that of the two hydrogen atoms at infinite separation: 
this is therefore a dissociative situation, and the molecule is 
unstable. The orbital responsible for this instability is referred 
to as an antibonding orbital. 

The case of two helium atoms being brought together is a 
good example of the effect of antibonding character. When 
the two nuclei are quite close together the Is-orbitals overlap 
appreciably: the amplitudes taken with the same phase inter- 
fere constructively in the internuclear region to give one 
"molecular orbital, and the amplitudes taken with opposite 



antisymmetry 



sign interfere destructively to give another, antibonding, 
molecular orbital. Four electrons have to be added to these 
two composite molecular orbitals: two enter the lower-energy 
orbital and tend to cause the nuclei to stick together; the next 
electron enters the antibonding orbital and so lowers the 
strength of the bond. The fourth "pairs with the third in the 
antibonding orbital, and the combined effect of this pair is 
sufficient to overcome the bonding of the first pair and to 
disrupt the molecule. Consequently He 2 is an unstable mole- 
cular species (even though HeJ is weakly stable in the gas 
phase). This description enables one to see why two helium 
atoms collide without sticking together: as the atoms approach 
the bond and antibond are formed, but both are occupied 
simultaneously and the rise in energy as the atoms approach 
appears as a repulsive force. It is quite easy to extend this de- 
scription to more complicated atoms and molecules and to 
understand why bulk matter is impenetrable. 

Questions. 1 . What effect does the occupation of an antibond- 
ing molecular orbital have on the energy of a molecule? Why 
does the overlap of atomic orbitals with opposite sign lead to 
an antibonding orbital? Why is HeJ less stable than He|* ( and 
He 2 unstable? Why, then, is the world not littered with He|*? 
Why do not two neon atoms stick together when they collide 
in a gas, and how may this argument be adapted to account 
for the same property of methane and N 2 ? Why is bulk matter 
impenetrable? 

2, Take a Is-orbital on each of two protons at a separation R, 
and use the mathematical form for the orbitals given on p. 275 
to plot the amplitude of the molecular orbitals that result 
when they are combined first with the same sign and then with 
opposite sign. Plot the electron density corresponding to two 
electrons in the bonding orbital and then to two in the anti- 
bonding orbital, and then plot the difference density (obtained 
by ignoring interference effects, calculating the electron 
density when each electron is confined to its own nucleus, and 
subtracting this density from the density calculated for the 
bonding and antibonding cases). Do this calculation for about 
three judiciously chosen nuclear separations, and reflect on 
the connexion of these results with the discussion in the text. 
A proper calculation should use "normalized orbitals, but a 



simple one is sufficient for illustration. What effect does 
normalization have on the difference densities? 

Further information. Antibonding effects are of considerable 
importance in determining molecular structure: see Coulson's 
The shape and structure of molecules (OCS 9) and Coulson 
(1961). A long essay on the importance of antibonding 
orbitals has been written by Orchin and Jaffe (1967). Further 
details will be found in MQM Chapter 9, and helpful advice on 
the calculation of overlap integrals is given by McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972) Chapter 2 and 
Appendix C. A compilation of molecular energies has been 
published by Richards, Walker, and Hinkley (1971). Difference 
density maps are given by Coulson (OCS 9) and by Deb 
(1973), and the latter gives references to many other sources. 

antisymmetry. A function f (x) is antisymmetrical (or 
antisymmetric) if f{— x) is equal to — fix). A wavefunction of a 
system containing N indistinguishable particles h antisym- 
metrical under particle interchange if it changes sign when the 
coordinates of any pair of particles are interchanged; that is, if 
^(ri,r 2 ) = ~"lM r J< r i )- Do not confuse the word with 
asymmetrical, which means the absence of symmetry. An 
example of the first type of antisymmetry is the function x, 
because (— x) = — (x); another example is the function sinx, 
because sin(— x) = —sin x. An example of an antisymmetrical 
wavefunction is i/' a (i'i)0' b (r 2 ) — (r 2 ) '/' b (r 1 ) because inter- 
changing the coordinates rj and r 2 , which by reference to 
Fig. A5 is seen to be equivalent to interchanging the particles, 
changes the sign of the function. 




FIG. A5 Interchanging particles 1 and 2 is equivalent to interchanging 
the vectors t\ and rj. 



8 



aromaticity 



The importance of antisym metrical wavefu net ions stems 
from the "Pauli principle, which demands that the total wave- 
function of any collection of electrons must be antisym- 
metrical. The formation of a "Slater determinant is one way 
of constructing a fully antisymmetrized function out of a 
collection of functions. The imposition of the requirement of 
antisymmetry on a wavefunction introduces important modi- 
fications to the energy: see "exchange energy. 

Questions. State the condition on f{x) for it to be an antisym- 
metrical function of x. Which of the following functions are 
antisymmetrically 2 , x 3 , 3x 2 — 2x 3 , cosx, tan x, expx, expx 2 , 
cosecx? Show that any asymmetric function F(x) can be 
expressed as the sum of an anti symmetrica I function and a 
symmetrical function. What is meant by an antisymmetrical 
wavefunction? Which of the following wavefu net tons are 
antisymmetrical: \p (r ( )i£ (r a ), sin [*{ri~r 2 )], 
*.(ri »* b ('a )* c ('3» - * B (r 2 )^ b (r, )^ e (r 3 ) + 
^ a (r 2 )^ b (r 3 f ^(rj ) — ...? Show that the last can be 
written as a 3 X 3 determinant. What is the importance of 
antisymmetry in quantum mechanics? 



Further information. See MQM Chapter 8 for the reasons lying 
behind the use of antisymmetric wavefuncttons. Helpful dis- 
cussions of the reasons and the consequences will be found in 
§29 of Pauling and Wilson (1S35), in §3-1 of Slater (1963), 
and §T2 and succeeding chapters of Richards and Horsley 
(1970). The symmetry and antisymmetry of functions can be 
envisaged, and treated, as a problem for "group theory; there- 
fore see MQM Chapter 5, Cotton (1963), Tinkham (1964), 
Bishop (1973), and Altmann (1962). The antisymmetrization 
of wavefunctions is a problem in statistics: see Pauling and 
Wilson (1935), Condon and Shortley (1963), Hamermesh 
(1962), and Judd (1963). This reading sequence will illustrate 
how a simple requirement may have consequences of un- 
bounded complexity. 

aromaticity. An aromatic molecule is cyclic, planar, and con- 
jugated (possessing alternating single and double bonds) but 
with a stability greater than would be expected for a molecule 
with so many double bonds. The extra stabilization is due to 
■resonance (in "valence-bond language) or derealization (in 
"molecular-orbital language), and the extra stabilization energy 



is called the "resonance energy or derealization energy. 
-Benzene is the archetype of such molecules. It has been found 
that aromatic molecules possess 4/J+2 electrons (the Hiicket 
4n+2rule), where n is an integer. Thus benzene has n = 1, and 
the simplest aromatic molecule of all, which was first prepared 
not long ago, is the cyclopropene cation (/? = 0). The large 
molecule [18] -annulene, consisting of 18 conjugated carbon 
atoms in a planar ring, is also aromatic {n = 4), 

The basis>of the An+2 rule may be understood by con- 
sidering the energy levels of cyclic hydrocarbons. From N 
atoms N molecular orbitals may be formed. The lowest 
energy (most strongly binding) orbital has no "nodes, all the 
others have nodes, and, except perhaps the uppermost, are 
doubly "degenerate. See, for instance. Fig. A1 on p. 2, (This 
two-foid degeneracy may be viewed as a consequence of the 
fact that an electron may run round the ring in either 
direction.) In the case of N being an even number the upper- 
most level is non-degenerate. Into the lowest energy level one 
may insert two electrons, and into each doubly degenerate 
pair one may insert four electrons. Closed-shell molecules 
therefore contain 4/J+2 electrons, where n is the number of 
fully occupied doubly degenerate levels. Achieving a closed 
shell is an energetically favourable situation, especially as one 
is normally obtained by adding no more electrons than are 
needed to complete the bonding molecular orbitals, and so 
leaving empty the energetically unfavourable "antibonding 
orbitals. 

It has also been found that some molecules of a conjugated 
double-bond nature show an enhanced instability: they 
contain 4n electrons, and include the cyclopropene anion 
(n = 1 ). This great instability gives rise to the name 
antiaromatic. 

Further information. See MQM Chapter 10, Coulson's The 
shape and structure of molecules (OCS 9), Coulson (1961 ), 
Streitweiser (1961), Salem (1966), and Pilar (1968). A simple 
account of the preparation and properties of a variety of 
aromatic and antiaromatic molecules has been given by 
Breslow (1972). The analysis of the 4n+2 rule may be carried 
fbrther by referring to Streitweiser (1961), Chapter 10; there 
will be found described the concept of pseudoaromaticity and 
Craig's rules. See Bergmann and Pullman (1971 ) for the pro- 



ceedings of a conference on aromaticity, pseudoaromaticity, 
and antiaromaticity, 

atomic orbital. An atomic orbital describes the distribution 
of an electron in an atom: it is the °wavef unction for an 
electron in an atom. The classical Rutherford and °Bohr 
theory of the hydrogen atom sought a model of its structure 
in terms of a trajectory of the electron about the nucleus, and 
so the atom was viewed as a central nucleus with an electron 
in one of a variety of orbits. The introduction of quantum 
mechanics, and in particular the impact of the "uncertainty 
principle, showed that the concept of trajectory was untenable 
on an atomic scale, and so an orbit could not be specified. 
Quantum mechanics replaced the precise trajectory, the orbit, 
of the electron by a distribution, an orbital. An atomic orbital 
is a function i^(r} of the coordinates of the electron, and, in 
accord with the Born interpretation of the 'wavef unction, the 
probability that the electron may be found in an infinitesimal 
volume element dr surrounding the point r is \p*{r)\p{r)<iT. It 
follows that, if the form of the atomic orbital is known, we 
are able to predict the electron density at any point in the 
atom. 

As an example, the electron in the ground state of the 
hydrogen atom is distributed in an atomic orbital (an s-orbitat) 
of the form exp (— r/a ), where a is a constant, the Bohr 
radius (5-29 X 10~ n m): this implies that the orbital is 
spherical (the function depends on r but not on or 4>). and 
the electron density at any point depends on exp (- 2r/a ). 
Therefore the density is greatest at the nucleus, and then 
declines exponentially with distance. Such an atomic orbital 
can be represented by a spherical boundary surface within 
which there is some probability, let us say 90 per cent, of 
discovering the electron. In this boundary -surf ace represen- 
tation, which is depicted in Fig. A6, the s-orbital is drawn as a 
sphere of an appropriate radius; but it must not be forgotten 
that there is some slight probability of discovering the electron 
at points well outside the boundary surface. 

It is normally sufficient in the discussion of atoms to con- 
fine attention to the s-, p-, d- r and occasionally the f-orbitals. 
Of these only the s-orbital is spherically symmetrical; the 
others have an increasingly pronounced angular dependence 
corresponding to the electron being concentrated in particular 



atomic orbital 9 

directions in space. The form of these atomic orbitals will be 
found in the discussion of the "hydrogen atom. 

The amplitude of an atomic orbital depends on the distance 
from the nucleus. It is reasonable to expect the amplitude to 
diminish to zero at large distances from the parent nucleus, 
and this is found to be so: when r is very large all orbitals 
decay exponentially. The decay is not in general exponential 
at all distances from the nucleus, for in most some incipient 
undulations occur at small radii. This behaviour is examined in 
detail in the case of the "hydrogen atom, and only one point 
need be emphasized here: only for s-orbitals does the 
amplitude not drop to zero at the nucleus itself (see Fig. A6). 




FIG. A6. Boundary surface capturing different proportions of the 
electron in the ground state of the hydrogen atom (in a 1 s-orbital), R 
denotes the covalent radius (30 pml. 



The number of lobes possessed by an atomic orbital 
(actually the number of angular modes) determines the 
"orbital angular momentum of an electron in the orbital. The 
mean curvature of the orbital, which is determined by the 
number of radial and angular nodes (because the more often 
the wave must pass through zero, the more sharply is it curved), 
determines the "kinetic energy of an electron that occupies It, 
and the mean inverse distance from the nucleus determines the 
potential energy in hydrogen and the attractive part of the 
potential energy in many-electron atoms. In many-electron 
atoms the energy is also influenced by the interelectronic 
repulsions, and these have the further effect of distorting the 
electron distributions from those in hydrogen-tike atoms; these 



10 



atomic spectra: a synopsis 



effects are calculated by "self -consistent field methods. 
Although true orbitals of many-electron atoms are complicated 
functions, it is possible to make a fair approximation to them 
by orbitals that have the same angular dependence as those of 
hydrogen, but whose radial dependence is determined by a set 
of simple rules: these are the "Slater orbitals. 

When atomic orbitals are represented by boundary surfaces 
it must be borne in mind that in many-electron atoms these 
are only a crude representation of the actual electron dis- 
tribution, and in fact do little more than designate regions of 
space where the orbital has appreciable amplitude and where 
the electron has a high probability of being found. 

Questions. What is an atomic orbital? What information does 
it contain about the distribution and properties of an electron 
that occupies it? How does an orbital differ from an orbit? 
What is meant by a boundary surface? Calculate from the 
wavef unction given in the text the radius of the boundary 
sphere which captures 50 per cent, 80 per cent, 90 per cent, 
and 99-99 per cent of the electron density in the Is-orbital of 
hydrogen (a = 53 pm, 0-53 A), What is the effect on the 
shape of atomic orbitals of the interelectronic repulsions in 
many electron atoms, and how may they be taken into 
account? Sketch the boundary surfaces for p-, d-, and 
f-orbitals by referring to the pictures in the article on the 
•hydrogen atom. What is the evidence that electrons are dis- 
tributed in atoms In the manner we have described? 

Further information. See MQM Chapter 8. The shape and 
significance of atomic orbitals are discussed in detail in 
Coulson (1961), Herzberg (1944), White (1934), Pauling and 
Wilson (1935), and Kauzmann (1957). Information about 
self-consistent field and Slater orbitals wili be found under 
the appropriate headings. The electronic occupation of atomic 
orbitals is determined by the 'aufbau process and the "Paul! 
principle. As well as determining the structure of atoms the 
atomic orbitals are the basis of descriptions of molecular 
structure: see "molecular orbitals and -linear combination of 
atomic orbitats. See also "wavef unction and "radial-distribution 
function. Read the section on the hydrogen atom for a 
detailed discussion of its orbitals. 



atomic spectra: a synopsis. An atomic absorption or 
emission spectrum arises when an atom makes a transition 
between two states (which are often called "terms): the 
combination principle states that all lines in a spectrum can 
be represented as the difference between two terms {for the 
word 'term' is also used to denote the energy of a term). The 
transitions observed can normally be ascribed to "electric 
dipole transitions, and the lines that may appear in the 
spectrum are governed by the "selection rules. Their intensity 
is determined by the magnitude of the transition dipole 
moment. The appearance of the spectrum may be modified 
by the application of a strong magnetic field (the "Zeeman 
effect) or a strong electric field (the "Stark effect), because 
both fields can cause small shifts in the energy levels of the 
atom. On the gross structure of the spectrum is seen a "fine 
structure, which is interpreted in terms of spin-orbit coupling, 
and an even finer "hyperfine structure which is due to the 
interaction of the electrons with the "magnetic dipole and 
electric "quadrupole moments of the nucleus. 

The information of chemical interest that can be obtained, 
or has been obtained, from the spectrum of an atom is as 
follows: 

1. Identification of species. Since every element has a 
characteristic spectrum, atomic spectroscopy m3y be used in 
analysis: the spectrum is used as a fingerprint for elements. 

2. Evidence for quantization. The study of the spectrum of 
atomic "hydrogen was of profound significance for developing 
the ideas of "quantum theory and quantization. Out of the 
study emerged the idea of "atomic orbitals, and all the other 
paraphernalia of quantum chemistry. 

3. The Pau/i principle. From a study of the spectrum of 
helium emerged the puzzling result that not all the states of 
the atom are allowed. The "Pauli principle was the rational- 
ization of these data, and its discovery was the key that 
enabled the periodic system to be explained. 

4. Atomic energy levels. The study of atomic spectra yields 
information on the energy levels of atoms: we are able to say 
how deeply electrons are buried in inner shells and which 
electrons and states of the atom are likely to be important in 



atomic units 



11 



governing the bonding properties of atoms {their valence, and 
the strength of the *bonds they form). We need to know the 
energy of atomic energy levels in order to assess the energy of 
the "valence state and the rote of "hybridization. Photo- 
chemistry depends on a knowledge of the energy levels of 
excited atoms. 

5, Ionization potentials. The energy required to ionize an 
atom (its "ionization potential) can be determined from atomic 
spectroscopy; so too, with more difficulty, can some "electron 
affinities. Both these properties are central to an understanding 
of the structure and reactions of atoms and molecules. See, for 
example, "electronegativity. 

6. Spin-orbit coupling. From the -fine structure we may 
determine the spin-orbit coupling constant. This is of use in 
the discussion of the role of triplet states in photochemistry 
(for example, the heavy atom effect in quenching "phos- 
phorescence) because it determines the rate of -singlet-triplet 
inter-system crossing. We also need to know spin-orbit 
coupling constants to evaluate -^-values in "electron spin 
resonance and to discuss the structure of molecules— see, for 
example, the -Hund coupling cases. 

1 . Hyperfine coupling constants. From the "hyperfine 
structure of spectra can be determined the strength of the 
magnetic and electric coupling of electrons to nuclei, and also 
the °spin of nuclei. Such coupling constants are important for 
the "hyperfine effect in "electron spin resonance and the 
"spin-spin coupling in "nuclear magnetic resonance. 

8, X-ray Spectra, Spectra in the short-wavelength °X-ray 
region were the basis for Moseley's determination of the 
atomic numbers of the elements. 

Further information. See MQM Chapter8foradescriptionof 
the interactions that lead to the structure of atomic spectra 
and a more detailed description of their form. Introductory 
books on atomic spectra include those by Whiffen (1972), 
Barrow (1962), Herzberg (1944), Woodgate (1970), and 
White (1934). A book with many examples and with compre- 
hensive coverage at a slightly more advanced level is that by 
Kuhn (1962). More detailed analysis will be found in Shore 
and Menzel (1968), Candler (1964), who gives much exper- 



imental data, Condon and Shortley (1963), Judd {1963, 
1967), and Wigner (19S9). The books by Condon and 
Shortley and Wigner are classics: the former was written 
before many of the impressive angular-momentum techniques 
were developed but has been a dominating influence on the 
development of the subject, and the tatter is a classic and 
original exposition of atomic structure and spectra in terms of 
symmetry and "group theory. Both books may be regarded as 
ancestors of Griffith (1964), who, after his description of free 
atom spectra, develops the theory of the spectra of atoms in 
complexes: see "crystal -fie Id theory. Data from atomic 
spectroscopy will be found in Moore (1949 et seq.), who lists 
energy levels. Applications to photochemistry are described by 
Wayne (1970) and Calvert and Pitts (1966). 

atomic units. The appearance of many equations in quantum 
mechanics may be considerably simplified if mass is expressed 
as a multiple of the electron mass m (so that the mass of 
the electron is taken to be unity); charge as a multiple of 
the proton's charge e; length as a multiple of the 'Bohr 
radius a ; and energy in multiples of twice the ionization 
potential of the ground state "hydrogen atom. {Twice the 
ionization potential, 27-21 eV, ore 2 /47re a , is generally 
employed, although some people use the energy itself; that is, 
13-65 eV.) A consequence of this choice of units is that h= 1. 
The units may be augmented by the choice c = 1 for the speed 
of light (and so all velocities are expressed as a fraction of the 
speed of light in a vacuum). The units so chosen eliminate 
many of the constants in the "Schrodinger equation, and the 
numbers that emerge for various properties can be translated 





BOX 1: Atomic units (a.u.) 




quantity value 


1 a.u. of 


mass m a 9-109 X ID" 31 kg 
length a ° 5-292 X 10" n m 
charge e 1-602 X 10" 19 C 
energy e 2 /4m g a 27-2 eV; 2625 kJ mof ' 
velocity c 2-998 X 10 8 m s" 1 


Consequently ft = 1 ; $i B = 5; ff H = j. 



12 



aufbau principle 



into conventional units by re-introducing the units of mass, 
length, charge, and energy; see Box 1. 

aufbau principle. The aufbau or building-up principle is the 
statement about how electrons should be fed into the orbitals 
of an atom or molecule in order to construct the species. The 
principle states that an electron enters the lowest available 
orbital consistent with the requirements of the "Pauli exclusion 
principle. This implies that the first electron enters the lowest 
orbital, the second joins the first (but with opposite "spin); 
the third electron enters the next higher orbital, then the 
fourth pairs with it and so on. See Fig. A7, When a set of 
"degenerate orbitals is to be filled {for example, when the p- 
shell of an atom is being populated) the first electrons enter 
different members of the set with parallel spin, in accord with 
the °Hund rule, and only when each degenerate orbital con- 
tains one electron do the remainder enter with paired spins. 
The list of the orbitals populated by the application of the 
aufbau process constitutes the "configuration of the atom. 




FIG. A7. The order of filling energy levels according to the aufbau 
principle. 



Questions, What is the aufbau principle? In what way does it 
depend on the Pauli principle? How would the principle 
differ if the particles being filled into the orbitals were 



"bosons? What is the role of Hund's rule? Discuss the structure 
of the atoms He, Li, Be, B, C, and N in the light of th^ aufbau 
principle, and also the structure of the molecules H 2 , N s , 2 , 
Fj, and NO (get help from "molecular orbitals). Discuss what 
might happen when two energy levels lie close together, but 
are not degenerate: under what circumstances might each be 
half-filled before the lower is filled? 

Further information. See MQM Chapter 8, Chapter 3 of 
Herzberg (1944), and Murrell, Kettle, and Tedder (1965). 
The aufbau principle is the basis of the periodic table; there- 
fore see how it is applied in Puddephatt's The periodic table of 
the elements (OCS 3) and in Chapter 2 of Phillips and Williams 
(1965). For the application of the principle to molecular 
systems see MQM Chapter 9, Coulson's The shape and 
structure of molecules {OCS 9), and Coulson (1961 ). The 
aufbau principle is important in transition-metal chemistry 
because in complexes the metal ion has a number of close-lying 
energy levels, and the situation in the last question is common: 
see "crystal -fie Id theory and "ligand-field theory. 

Auger effect. Other names for this effect are auto-ionization 
and pre-ionization. The Auger effect is a radiationiess tran- 
sition (a transition between states that involves no emission or 
absorption of radiation) from an excited state into a dis- 
sociative state. Consider an atomic energy level scheme of the 
type shown in Fig. AS: in this atom one series of levels 
terminates {at an ionization limit) at a lower energy than the 
other. Suppose we monitor the absorption spectrum of the 
atom, and we concentrate on the series on the left (l_>. The 
spectrum observed consists of a series of sharp lines at increas- 
ing frequencies, but when the frequency corresponds to an 
energy above the ionization limit of the other series (R), for 
example when we observe the transition to the line A, a 
marked change appears in the spectrum. The most noticable 
difference is in the sharpness of the lines, for A,B ... are 
blurred. They may also be slightly shifted. What is happening 
is that some "perturbation in the atom (for example, the spin- 
orbit coupling) is mixing the states of series L with those of 
series R, and therefore the states of L take a little of the 
character of R. But above the ionization limit of R this implies 
that the states of L have a tendency to ionize, the tendency 



Auger effect 



13 






\orta&tQ$ limit of I 




rodiotjanfcu 
transitions 



i8sia3tian--(ni|.of, R 



spectrum end harwtions of L 



FIG. A8. The Auger effect: the broadening of the spectrum (on the left) 
occurs where the atom makes radiationless transitions from states on the 
teft to the unbound states on the right. 



increasing as the contamination increases. Since the lifetime of 
the levels A, B, ... is diminished by this mixing, the width of 
the levels is increased ("uncertainty principle). The mixing of 
states R into states L may be expressed as a probability that an 
atom in a state of L makes an actual transition into a state of 
R, and the Auger effect is simply this type of radiationless 
transition into a dissociative state. The name pre-ionization 
reflects the fact that ionization occurs in series L before, on 
energetic grounds, it is expected, and the name auto- ionization 
reflects the 'self-induced' nature of the process in the sense that 



the perturbations within the molecule induce the ionization by 
flipping the bound state into the ionizing state. 

The Auger effect was originally detected in "X-ray 
spectroscopy, where the bombardment of a solid with fast 
electrons excites a K-sheli electron (let us say), and an X-ray 
is emitted when an L-shell electron falls into the vacant hole. 
A competing process is introduced by the Auger effect, 
because the excitation of the K-electron may induce auto- 
ionization, and another electron is boiled off the atom or out 
of the solid. The ionization process competes with the forma- 
tion of X-rays and diminishes their intensities. The effect is 
not wholly bad for, if the energy of the Auger electron is 
measured, information may be obtained about the energy 
levels of electrons in solids. This is the basis of Auger spec- 
troscopy. 

For the sake of completeness. Auger is pronounced 036. 

Questions. What is the Auger effect? How may it be detected 
in atomic spectra? Why do spectral lines become broadened by 
virtue of the Auger effect? What perturbations may cause auto- 
ionization? Describe the appearance of the spectrum showing 
pre-ionization. What is the role of the Auger effect in X-ray 
spectra? What effect does it have on the lines? Suppose we 
were looking at an atomic emission spectrum, what would be 
the influence of the Auger effect? In what sense may "predis- 
sociation of molecules be considered to be an Auger effect? 
What other processes can you think of that, however loosely, 
may be considered to be the manifestation of an Auger effect? 

Further information. An account of the Auger effect in atomic 
spectra will be found in §4.2 of Herzberg {1944) and §4.5 and 
§5.3 of Kuhn (1962). The rote it plays in the formation and 
appearance of X-ray spectra is described by Burhop (1952) 
and has been reviewed by Burhop and Asaad (1972). For a 
discussion of Auger spectroscopy, see Siegbahn (1973) and 
further references under "ionization potential. See also 
"predissociation. 



B 



band theory of metals. The theoretical description of the 
structure and properties of metals is based on the view that in 
gross terms they are composed of an array of positive ions held 
together and surrounded by a sea of electrons. The properties 
of this sea of electrons determine the typical characteristics of 
metals: their electrical conductivity, thermal conductivity, 
reflectivity, malleability, and ductility. The energy levels 
available to the electrons are of paramount importance, and 



forbidden 
enemies 



permitted 
energies 



4- 



forbidden 
eneiqies 



permitted 
eneiqie* 




T 

r mm 



FIG. B1 . The extent of occupation of bands separated by gaps deter- 
mines whether a material is a conductor or an insulator. Occupied levels 
are coloured. 



analysis of the problem shows that the available energies fall 
into bands, and that between these bands lie regions of energy 
which no electron can possess. If a band is less than full 
(Figs. B1a and c), the electrons in it can be induced to move 
under the influence of a small disturbance: hence the high 
conductivity of metals. If the bands are full, electrons can be 
induced to move only if they can be supplied with enough 
energy to excite them through the forbidden band gap into 
an empty upper band: this is an energetically unfavourable 
situation, and so such materials are electrical insulators 
(Fig. B1b). 

The formation of energy bands in metals may be described 
in a number of ways: we shall discuss two. The first starts 
from the view that a metal may be envisaged as a massive 
molecule, a molecule of almost infinite extent. The molecule 
is composed of atoms in a regular lattice, and on these atoms 
there are "atomic orbitals which overlap with their neighbours' 
atomic orbitals. This situation of overlapping orbitals is 
encountered on a much smaller scale in conventional molecules, 
for it is the basis of the "molecular-orbital description of mole- 
cular structure. Let us pretend that only a single s-orbital is 
available on each atom, and for simplicity we consider a linear 
crystal (a single line of N atoms}. 

Concentrate first on two of the atoms in the chain: their 
orbitals 'overlap and form a bonding and an antibonding pair 
of molecular orbitals with an energy separation determined 
largely by the amount of overlap (Fig. 82], Now bring up a 
third atom to a lattice distance and let it overlap with its 
immediate neighbour (and for simplicity, and as a good ap- 
proximation, neglect its small overlap with its next- nearest 



14 



band theory of metals 



15 



-c 




-m 




N~oo 




FIG. 82, As 2, 3, 4, ... N atoms are added in a line the energy levels 
begin to fill a band; when N is very large the energy levels with in the 
band virtually form a continuum. 



neighbour). This adjusts the molecular orbitals and their 
energies so that three orbitals are obtained in the place of the 
former two; one is bonding, one is nonbonding r and the third 
is antibonding. This is illustrated in Fig. B2. The process of 
sliding an atom along the line up to a lattice point [rather like 
using an abacus) may be continued, and Figs. B2 and B3 
show that the effect is gradually to fill in the energies until 
when N is very large the N energy levels form a virtually con- 
tinuous band of energies of width A , Since the band has been 
formed from s-orbitals it is called the s-band. 

If each atom contributes one electron (the atoms might be 
sodium atoms in a line), then since each energy level consti- 
tuting the band may accommodate two electrons ("Pauli 
principle), in the metal half the band will be full of electrons, 
and the upper half will be empty, as in Fig. BTa. (This is just 



the application of the °aufbau process to a collection of energy 
levels.) The presence of the half-filled band means that the line 
of atoms behaves as a metal. If instead each atom were to con- 
tribute two electrons, the band would be filled, as in Fig. B1b, 
and an applied electric field would be unable to shift the 
electrons: the line of atoms is then an insulator. Why motion 
cannot occur in this case will be described below in more 
detail. 

It is improper to restrict the formation of a band to the 
overlapping of s-orbitals. If the metal atoms have p-orbitals in 
their valence (outermost) shell these too should be allowed to 
overlap. Because of -shielding and penetration effect, p-orbitals 
lie higher in energy than s-orbitals, and so their overlap gives 
rise to a band (the p-bancl) above the s-band, and separated 
from it by a gap, the magnitude of which depends on the 
strength of bonding between the atoms and the s-p separation 
of the isolated atoms (Fig. B4). Here we see the reason for the 
appearance of the bands of energies, and the regions of 
forbidden energies. 

The approach to the band structure just described is some- 
times referred to as the tight-binding approximation (TBA) 
because it takes the view that electrons tend to stick to nuclei, 
and therefore are quite well described by atomic orbitals 
characteristic of the atoms. An alternative approach starts 
from the view that a good first approximation is to suppose 
that the electrons have no electrostatic interactions and are 
completely free to swim about in the bulk of the metal, and 
that the presence of a periodic lattice may be imposed as an 
improvement in a second step: this is the nearly -free electron 
approximation (NFE). 

How does this alternative approach lead to band formation? 
Let us once again consider the one-dimensional metal, but 
begin by ignoring the presence of positive ions. The situation is 
now that of a collection of 'particles in a box, or, since the 
line is of almost infinite length, the same as the free-electron 
system. The energies available to free electrons take a con- 
tinuous range of values and depend on the momentum ±kh 
according to k 7 (Fig. B5a). Pouring electrons into this con- 
tinuum of energy levels fills them up to some energy E , the 
Fermi energy, and leaves vacant all the energy levels above. 
The filled levels constitute the Fermi sea, whose surface, the 
Fermi surface, is at E p . On this model everything would be a 



16 band theory of metals 

Enenjy 




FIG. 63. The actual snergv levels 
for a chain of N atoms, each inter- 
acting only with its nearest 
neighbour. Note that by N - 20 
the band is getting dense, and that 
its width does not expand indefi- 
nitely. 



6 7 8 9 

Number of otorrn in row 



II 12 20 




! p-band 




s-p separation in oroms 

band qap in metal 




FIG. B4. Band-gap formation, s- and p-bands, and the orbital overlaps 
corresponding to the extreme energies of each band. 



metal, for the 'band' has no upper limit, and so can never be 
filled. This emphasizes the central role that the periodic lattice 
plays in determining whether or not a material is a metal. 

In order to see how the lattice imposes a band structure we 
remind ourselves that the states of the free electron are waves 
of definite length (wavelength = 2ir/k), and the true system 
may be regarded in terms of these waves propagating through 
the periodic lattice. At wavelengths long compared with the 
lattice spacing a the waves slide through, but when the wave- 
length is comparable to a the lattice diffracts the waves. When 
the wavelength is equal to twice the lattice spacing a pattern of 
standing waves is set up because a wave that begins moving to 
the right is reflected by the lattice and moves to the left, 
where it is reflected to the right, and so on. These standing 
waves, with wavelengths in the vicinity of 2a, have a stationary 
distribution in space, and we may envisage two types of 
arrangement. In one arrangement the standing wave has its 
amplitude maxima at the positions of the lattice points 
(positive ions), and in the other the maxima are between the 
lattice points. Whereas in the free-electron model these stand- 



band theory of metals 



17 



second 
Briiiouin lone 




first 
BfffJouin zone 



FIG. B5. (al The free-electron energy levels and (b) the band structure 
imposed by the periodic lattice. Both contain a number of electrons 
that makes them metals. 



ing waves had the same energy, the presence of the periodic 
lattice implies that they have different potential energies, and 
so we discover that at k = ±ir/a there is an energy gap (see 
Fig. B6). Another gap is found at k = ±2n/a, and so on; there- 
fore, we see that the periodic lattice splits the free-electron 
continuum of energy levels into a series of bands separated by 



gaps (Fig. 65). Into these bands, the Briiiouin zones, may be 

inserted the electrons of the metal, and metallic properties 
are predicted only if there remain incompletely filled bands. 

The relation of the NFE bands to the tight-binding bands 
may be discovered by comparing the form of the orbitals at 
the edge of the s- and p-bands with the waves at the edges of 
the first and second Briiiouin zones. The immediate obser- 
vation, from Fig. B6, is that the noda! structure is the same; 
only the details of the electron distributions differ, especially 
in the regions close to the nuclei. 

Well below the Briiiouin zone boundary (that is, for energy 
levels for which \k I < ir/a in the first zone) the states of the 
electron may be regarded as running waves {as opposed to the 
standing waves forced on us at the zone edge). In one dimen- 
sion the waves run with momentum +kh to the right and —kh 
to the left; since they have energy proportional to k 2 these 
two running waves are "degenerate (have the same energy). 




FIG. B6. Energy-gap formation due to a periodic lattice and its relation 
to the orbital (TBA) approach (in colour). Note that the uppermost 
level of the lower band and the lowermost of the upper have the same 
wavelength but one has nodes at the nuclei, the other between them. 
The lattice spacing is a. 



18 



band theory of metals 



Therefore, in a metal there are equal numbers of electrons 
running to the left and to the right, and in the absence of 
external stimuli no current flows. When an electric potential 
difference is applied, the energy of the electrons running to the 
right may differ from those running to the left (Fig, B7) and 
so the electrons redistribute themselves in order to attain the 
lowest energy: this means that more occupy states corre- 
sponding to the flow of electrons to the right, because a lower 
energy is attained thereby, and the equilibrium situation shows 
a steady flow of current. If however the band is full the elec- 
trons cannot reorganize themselves to give a net direction to 
travel (consider Fig, B7 with a full band), and so no current 
flows. Such a material is an insulator. A really enormous field 
might so affect the energies that an empty zone is brought 
down to the filled one: in this case of dielectric breakdown a 
current may flow, but often at a cost of disrupting the 
material. 




k— ft/a 



FIG. B7. When a suitable potential difference is applied to a material 
with an unfilled band the lowest energy distribution of electrons has 
more moving to the right [positive k) than to the left (negative k). 



The band theory of metals is able to account for the 
thermal properties of metals. For example, the high thermal 
conductivity may be traced to the way in which an electron 
carries heat rapidly through the lattice. Applying heat to one 



end of a metal rod induces lattice vibrations: these lattice 
vibrations excite an electron with an energy close to the Fermi 
surface into an unfilled state just above the sea, and so it may 
skim through the lattice with its high energy. Sooner or later it 
will plunge back into the Fermi sea, and in the process will 
impart its excess energy to the lattice, but this part of the 
lattice might be some way from the heated end of the rod. 
Thus heat is conducted through the rod. 

Pioneers of the theory of heat capacities of metals were 
worried by the presence of a great sea of electrons, all of which 
ought to be able to contribute to the heat capacity of a block 
of metal. Band theory dispelled this worry by drawing atten- 
tion to the fact that when a metal is excited thermally an 
energy of the order of kT\s supplied, and only those electrons 
within an energy kT of the Fermi surface can be excited into 
an empty level by such a stimulus. Consequently only a very 
small number of electrons are able to contribute to the heat 
capacity of a metal. 

Questions. 1. A metal may be pictured as an array of positive 
ions embedded in a sea of electrons: how is this picture able to 
account for the malleability and ductility of metals? What 
other characteristics of metals must any theory of their struc- 
ture explain? What is the role of energy bands in determining 
electrical and thermal conductivity? Why are clean, smooth 
metal surfaces highly reflecting? Describe the tight-binding 
approximation, and explain why bands may be formed. What 
is the difference between s- and p-bands? Consider a one- 
dimensional metal lattice, and let there be s- and p-orbitals in 
the valence shell. Describe the number of electrons that would 
give rise to metallic or insulating properties to the system 
(consider only p-orbitals along the tine). What determines the 
width of the bands? Guess whether an s- or a p-band is the 
wider. Discuss the thermal conductivity of the lattice you 
have just considered. What is the nearly-free electron approxi- 
mation? Why does the presence of a periodic lattice introduce 
discontinuities into the energy versus k dependence? Discuss 
how these bands are related to the tight-binding bands. What 
is a Brillouin zone? Discuss the conditions for a material to be 
a metal on the NFE theory. What is meant by the terms 'Fermi 
energy', 'Fermi surface', and 'Fermi sea'? What explanation of 
the heat capacity of metals is provided by the band theory? 



benzene 



19 



Suppose that an empty band is within an energy kT of the 
upper edge of a completely filled band: discuss the depen- 
dence of the electrical properties on the temperature of this 
semiconductor, 

2. Treat the one-dimensional chain of N atoms as a "Htickel 
problem and show that as atoms are added to the chain the 
levels gradually form a continuous band, but of finite width. 
Proceed by setting up a "secular determinant based on a 
resonance integral jS between neighbours, and between non- 
neighbours. Let there be N atoms in the chain. Show that the 

roots of the equation lie at 2|3eosl/J7T/(rt-rT )] , n = 1 , 2 /V, 

and plot the energy levels for N = 1, 2, 3, . . . 12, °°. 

Further information. A simple account of the structure of 
metals and insulators and the way the theory may be used to 
account for a wide range of properties is given in Chapters 6 
and 7 of Solymar and Walsh (1970) and in Jennings and 
Morris, Atoms in contact (OPS 4). Then see Altmann (1970) 
for a more quantitative account, but still at a moderately 
elementary level, and Quinn (1973), who gives an account of 
the TBA and NFE approximations and their more sophisti- 
cated developments. See also Kitte! (1971), Ziman (1972), 
and Dekker (1960). Semiconductors are simply described in 
Solymar and Walsh (1970). Amorphous materials are of 
immense technological importance, and a summary of this 
difficult field has been given by Mott and Davis (1971 (. 

benzene. Benzene is the archetype of the "aromatic mole- 
cules in which the ring of carbon atoms is unusually stable 
when compared with other unsaturated systems. The -reson- 
ance theory of chemical structure is the -vale nee- bond (VB) 
attempt to explain this stability, and the derealization picture 
is the -molecular-orbital (MO) attempt. 

The classical picture of the structure of benzene is that of 
a hexagon of alternating single and double bonds. The energy 
of this Kekute structure may be calculated by taking into 
account all the electrostatic interactions of the electrons and 
nuclei. But the interelectmnic repulsions have the further 
effect of pushing the ir-electrons into the vacancies between 
the double bonds; that is, there is a tendency for the molecule 
to turn itself into the other Kekule structure (Fig, B8). If the 
transformation is not allowed the molecule has an energy E ; if 




FIG. B8. Some of the electrostatic repulsions are relaxed by permitting 
resonance between two Kekule' forms. 



the transformation is allowed the energy falls below E because 
the structure is more relaxed. The "resonance picture of the 
stabilization may therefore be interpreted in terms of the 
lowering of energy brought about by permitting the molecule 
to 'resonate' between the two Kekule structures. This picture 
of two resonating Kekule structures loses its edge when it is 
realized that these two structures contribute only about 
80 per cent of the true total structure (which is a "super- 
position of resonating structures), the remaining 20 per cent 
being due largely to Dewar structures (Fig, B9). The 'reson- 
ance' picture gets rather muddy at these depths, although it 



Conventional representation 



Kekule form f 80/ ) 



Dewar form 



C20 % J 




FIG. B9. Five resonating forms contributing to the VB structure of 
benzene. The true structure, designated by its conventional represen- 
tation, is a superposition principally of these fundamental structures. 



20 



benzene 



is valid when treated properly, and further details are given 
under "resonance. 

The "molecular-orbital theory gives the impression of 
providing a more transparent description of the stability of 
the ring, based on the fact that benzene is an even "alternant 
molecule which may be described in terms of the orbitals and 
energies shown in Fig. B10. Six electrons are to be added to 




FIG. B10. Hiickei energy levels and orbitals for benzene. The numbers 
are the coefficients of 2p for each atom; the sign of the coefficient is 
marked on the orbital. Obtain unmarked coefficients by symmetry. 



the del oca I i zed 7r-system, and, as the first three orbitals are 
bonding, all the added electrons contribute to the stability of 
the molecule. 

The 0"-electrons have an important role to play in both the 
VB and the MO descriptions because six carbon atoms sp - 
"hybridized (with lobes at 120°) form a six-membered ring 
(with angle 120°) without strain. Each of the six carbon atoms 
has its full share of its valence electrons, and so not only is 
there no energy arising from an imbalance of charge, but there 
are also no centres of charge excess or deficiency to provide a 
reactive site (see "alternant hydrocarbon). The hexagonal ring 
of carbon atoms is a very well poised system. 

Questions, 1. Why does the "resonance of two Kekule' struc- 
tures lend stability to the benzene ring? What structures must 
be taken into account in the full vatence-bond treatment, and 
what proportion do they constitute of the total structure? 
Should ionic structures be included too? Draw some. What 
features of the hexagonal structure of benzene make the 
molecule stable in molecuiar-orbita! terms? What features of 
this explanation are equally applicable to a valence-bond 
description? 

2. Find combinations of the three occupied benzene molecular 
orbitals that may be interpreted as the three localized bonds 
of a Kekule structure. (See -localized orbitals.) How may the 
other structure be obtained from the molecular orbitals? How 
may the Dewar structures be obtained? 

3. Use the symmetry of the molecule to find the molecular 
orbitals in the Huckel scheme, and find their energies. Find 
the "resonance energy of the molecule. The experimental 
value is 150 kJ mof 1 (how would you determine this 
experimentally?). 

Further information. See MOM Chapter 9 for the simple 
molecuiar-orbita! description of benzene. A simple account 
of the structure of benzene is given by Co u I son in his The 
shape and structure of molecules (OCS 9) and in Chapter 9 of 
Coulson (1961). For the valence-bond description see also 
Pauling (1960), For more mathematical details of the cal- 
culations see McGlynn, Vanquickenborne, Kinoshtta, and 
Carroll (1972), Salem (1966), and Pilar {1968}. For a recent 
review of the spectroscopy of benzene see Murrell (1971) and 
Hall and Amos (1969). Other entries of interest are "resonance, 



birefringence: a synopsis 



21 



"aromaticity, "Hu'ckel method, and "alternant. Some physical 
properties of benzene are collected in Table 1. 

birefringence: a synopsis. Birefringence denotes the 
presence of different "refractive indexes for the two polar- 
ization components of a beam of light. Birefringence may be a 
natural property of isolated molecules, or of a particular 
crystal form, or it may be induced electrically, magnetically, 
or mechanically. An important example of natural birefring- 
ence is optical activity in which the plane of polarization of 
light is rotated as it passes through the medium. This is an 
example of circular birefringence, because the effect depends 
on the different rates of passage of the left and right circu- 
larly polarized components (remember that the velocity of 
light in a medium hc/n). The Faraday effect is the induction 
of circular birefringence in a naturally inactive sample by the 
imposition of a longitudinal magnetic field. The rotation 
induced is proportional to the field strength, and the constant 
of proportionality is the Verdet constant; all molecules give a 
Faraday rotation. A magnetic field applied transversely induces 
a linear birefringence in which one plane-polarized component 
travels faster than the other: this induces an ellipticity into the 
beam, and is referred to as the Cotton-Mouton effect. The 
same phenomenon induced by a transverse electric field is 
called the Kerr effect Because the "refractive index and the 
absorption coefficient are related (by the Kramers- Kronig 
dispersion relations), and when one depends on the polar- 
ization so does the other, both the Kerr and the Cotton- 
Mouton effects may be observed by monitoring the different 
absorption coefficients for the polarization components. 
Birefringence may be induced by fluid flow if the molecules 
are sufficiently anisotropic: the alignment of long thin mole- 
cules introduces an anisotropy into the optical properties of 
the medium and the refractive indexes depend on the orien- 
tation of the polarization of the beam: this is streaming 
birefringence. 

Further information. See MQM Chapter 1 1 for a discussion of 
the quantum-mechanical basis of natural optical activity. A 
further simple discussion has been given by Kauzmann (1957), 
and a thorough review, which also deals with induced birefrin- 
gence, is in the book by Caldwell and Eyring (1971). See 



Fredericq and Houssier (1973) for a simple account of electric 
drchroism and electric birefringence, especially their appli- 
cations. Other reviews containing interesting material are those 
of Moscowitz (1962), Tinoco (1962), Mason (1963), and Urry 
(1968). For a further discussion of the Faraday and Cotton- 
Mouton effects, see Buckingham and Stephens (1966). For the 
application of induced birefringence in liquid crystals to dis- 
play devices, see the article by Elliott (1973). For the optical 
properties of solids, see the standard work on optics by Born 
and Wolf (1970), Landau and Lifshitz (1958a), and Wooster 
(1973). The dispersion (the frequency dependence) of optical 
activities is a basic research tool for the study of the stereo- 
chemistry of molecules: for an introduction to optical 
rotatory dispersion (ORD), see Crabbe (1965). 

black-body radiation. A black body is one that absorbs all 
the radiation incident upon it. A practical example is a con- 
tainer completely sealed except for a tiny pinhole: this hole 
behaves as a black body because all light incident on it from 
outside passes through and, once in, cannot escape through 
the vanishingly small hole. Inside it experiences an indefinitely 
large number of reflections before it is absorbed, and these 
reflections have the result that the radiation comes into 
thermal equilibrium with the walls. Within the cavity we can 
imagine the electromagnetic field as having a distribution of 
frequencies characteristic of the temperature of the waits. The 
presence of the hole enables a small proportion of this equi- 
librium radiation to seep out and be detected, and the distri- 
bution of wavelengths in the black-body radiation is the same 
as the distribution within the equilibrium enclosure because 
the pinhole is a negligible perturbation. Black-body radiation is 
the radiation in equilibrium with matter at a particular temper- 
ature. 

The general features of the distribution of frequencies are 
familiar from everyday experience: at low temperatures the 
hole does indeed look black, but a sensitive detector would 
show that a small amount of long wavelength radiation is 
present. At higher temperatures the amount of energy 
emitted is much greater, and its principal component lies in 
the infrared. At still higher temperatures the pinhole glows 
dull red, then white, and afterwards blue, and the total 
amount of energy radiated increases dramatically. As the 



22 



black -body radiation 



temperature is raised further the peak of radiation passes 
through the ultraviolet, although the temperatures at which 
this occurs are too great to be conveniently accessible, There- 
fore we have made two observations: that the energy present 
in equilibrium increases dramatically as the temperature is 
raised, and that the wavelength of the light shifts towards 
the blue and beyond. The former observation is summarized 
by Stefan's Law, that the total energy emitted at a tem- 
perature T is equal to al* ; the constant a is Stefan's constant 
and has the value 5-67 X 1 0" 8 W m' 2 K" 4 . The second 



400 

blue 



'WO V s " wo 
yellow red 



b 00 

infrared 



5000 K 




~~ i — i 1 — n — r~ 

0-2 04_ Wl 0-6 



MT/he 



FIG. Bit. Radiation density from a hot black body at different 
frequencies and temperatures. 



observation is summarized by Wien's displacement law, that 
the wavelength of maximum intensity is inversely proportional 
to the temperature: A T= 2-9 mm K. 

r m ax 

The notoriety of black-body radiation lies in its defeat of 
classical mechanics, and in its role in the inception of quantum 
theory. The Rayleigh-Jeans law was the result of applying the 
ideas of classical mechanics: Rayleigh, with Jeans's help, 
counted the number of oscillators of a particular wavelength 
that could be found in a cavity, and then applied °equipar- 
tition to associate an energy of k T with each mode of 
oscillation; the energy density at the wavelength A was pre- 
dicted to obey the rule 8rrkT/\ 4 (Fig. B11). This conforms 
with none of the observations mentioned above: it predicts 
an enormous energy density at short wavelengths (called, by 
Ehrenfest, the ultraviolet catastrophe) and an infinite energy 
in any enclosure (the integral of the energy over all wave- 
lengths diverges). There is no maximum in the distribution, 
except the absurd one at zero wavelength. According to 
classical mechanics, even a glow-worm would devastate the 
surrounding countryside with its radiation. 

Planck's quantum hypothesis was a device that saved the 
situation even though in the process it abolished classical 
mechanics. From thermodynamic arguments Pianck was 
forced to the conclusion that he needed a single distribution 
law that yielded the Rayleigh-Jeans formula at low frequencies, 
and the Wien displacement law at high frequencies; therefore 
he found a formula which when approximated at low and 
high frequencies gave the two taws (see Box 2). On reflecting 
on the significance of his formula he was forced to the con- 
clusion that it represented a situation in which a radiation 
mode of a specified frequency could not possess an arbitrarily 
small energy: a mode could be excited only in discrete 
amounts of energy, or quanta. Whereas the classical law said 
that if there is a mode of frequency v in the container, then 
with it should be associated its due portion of energy accord- 
ing to equipartion, namely kT, the quantum law says that, if 
we have that mode, it shall be excited only if It is supplied 
with at least its due amount of energy, namely hv. The con- 
stant h is now referred to as Planck's constant (66256 X 
10 -34 J s). This means that the very high frequency modes 
are not excited at normal temperatures because the thermal 
energy present in the walls of the container is insufficient to 



black-body radiation 



73 



pM 



BOX 2: Planck distribution law for black-body radiation 

Energy density {energy per unit volume) in the range dX at 
the wavelength X: 

dU(K) = p(X)dX 

nt \\- (MiA j expt-hcfSkT) 
PW \ X s j ] ^x 9 l-ficf\kT)\ ■ 

Energy density in the range dv at the frequency V. 
dU(v) = p(v)dv 

( sitfilA j exp(-ftf/frr) | 
\ c 3 ) jl-expl-fiWttTlf ' 

Rayfeigh-Jeans law (long-wavelength, low-frequency limits 
of above): 

p(X)~8nk77X 4 \>hclkT 

piv) ~ 8m> 2 * 77c 3 v < *T//j. 
Stefan's taw (total energy density <* T 4 ): 

y = ^ dXo(X) = JTdvp(f) = io/c)T* 

O = 5-6697 X 10" 8 W m" 1 K~" . 

Wien '$ displacement law (X_ T = constant) : 

*^ max 

X T=hclbk = b 

max 

6= 2-8978 X 10" 3 m K. 



supply them with adequate energy (see -quantum). This 
damping effect on the high-frequency oscillators quenches the 
rise of the Rayleigh-Jeans distribution at high frequencies 
(short wavelengths), and so it eliminates the ultraviolet 
catastrophe. Furthermore, it introduces a maximum into the 
energy distribution versus wavelength curve, and this is in 
accord with the Wien law at high frequencies (and the Wien 
constant for the maximum is found to be equal to hc/bk). 
Because of the elimination of the high-frequency excitations 
the total energy emitted at a temperature T is finite, and the 
Stefan-law dependence on T 4 is reproduced. Thus we are 
forced to accept the propriety of Ptanck's quantum hypoth- 
esis; and, since the interaction of matter and radiation is such a 
fundamental process, it should be no surprise that the ramifi- 
cations of the hypothesis affected the whole of our appreci- 
ation of the nature of the world. 



Questions. 1 . What is a black body, and how may it be 
realized experimentally? Why is the radiation emitted through 
a pinhole in an otherwise closed cavity of fundamental 
importance? What are the changes in the frequency distri- 
bution of light emitted by a black body as its temperature is 
raised? State Stefan's law and Wien's law. What is the basis of 
the classical calculation of black-body radiation, and why is its 
form unacceptable? What was Planck's contribution, and what 
is the effect of quantization on the high-frequency oscillators? 
Discuss the differences between the excitation of a classical 
and a quantum "harmonic oscillator. Calculate the high- 
frequency form of the Planck distribution law (Box 2) and the 
short-wavelength form. From the latter show that the distri- 
bution passes through a maximum inversely proportional to 7", 
and deduce an expression for the Wien constant, evaluating it 
numerically. Calculate the energy density radiated by a black 
body at wavelengths of 1 cm, 55 nm, and 200 nm when it is 
heated to 300 K, 1000 K, and 10 s K, Suppose that the fila- 
ment of an incandescent lamp is a black-body radiator (the 
approximation is not absurd) and calculate the temperature to 
which it must be raised to emit light predominantly in the 
visible region of the spectrum. 

2. This question concerns the determination of the distri- 
bution function, and falls into two parts; the first (which we 
shall do) inserts the quantum hypothesis into the Boltzmann 
distribution to get the mean energy of a mode, and the 
second (which we shall not do) counts the number of modes 
of a particular frequency. If the energy of an oscillator is 
confined to the values nhv, then the probability that the 
oscillator has an energy nhv is, according to Boltzmann, 
ex.p(—nhv/kT)/Z, where Z is the partition function, or 
^n»o exp l — nhvlkT) . The mean energy is the sum of 
[(nhv)exp(— nhvikT)IZ] over all the values of n (n — 0, 1, 
2, . . .). First evaluate Z by realizing that it can be written as 
the sum over x", with x suitably chosen, and so it is a geo- 
metric progression; and then evaluate the sum over the 
numerator by realizing that it can be related to dZ/dT. Hence 
find the mean energy of a mode of wavelength X = civ. The 
number of modes of radiation in the range dX at the wave- 
length X is 8"JTdXA 4 ,' hence find the Planck distribution law. 
The answer is quoted in Box 2. 



7A 



Bohr atom 



Further information. See MQM Chapter 1 for a derivation of 
the Planck distribution and the number of modes in a con- 
tainer. For the latter also see p. 41 of Heitler (1954), §2,1 of 
Power (1964), §1.4 of Bohm (1951), and p. 144 of Lin 
(1967). Black-body radiation is discussed in an historical 
perspective by Jammer (1966), and a recent review is that of 
Lin (1967). See also Ingram's Radiation and quantum 
physics (OPS 2) and §9.13 of Re if (1965) for a useful discussion. 
Numerical values of the Planck distribution over a wide range 
of temperatures and wavelengths, together with integrated 
intensities, will be found in the American Institute of Physics 
handbook, Gray (1972), p. 6.198 and in Abramowitz and 
Stegun (1965). 



Bohr atom. Before the discovery of quantum mechanics, 
Bohr applied the principle of quantization to the problem of 
the structure of the "hydrogen atom. He asserted that: 

(1) an electron remained in a stationary state until it made 
a transition; 

(2) a transition from a stationary state of energy E . to 
another of energy £ was accompanied by the emission or 
absorption of radiation with a frequency v determined by the 
condition hv = E — E . (this assertion is the Bohr frequency 
condition); 

(3) the permitted stationary states were to be found by 
balancing the nuclear electrostatic attractive force against the 
centrifugal effect of the angular momentum of the electron in 
its orbit. 

The quantum condition was imposed at the last stage, for 
Bohr asserted that 

(4) the only angular momenta permitted were those whose 
magnitude was an integral multiple of h. 

The calculation of the energy levels done on the basis of 
these postulates led to an expression of the form E = —Rln 2 , 
where R is the ■ Rydberg constant and n is the principal 
"quantum number {n = 1 , 2, . . .). This expression is in 
virtually exact agreement with experiment, and was cause for 
jubilation. 

Refinement of this promising model proceeded in three 
steps. The first took into account the fact that the orbital 
motion occurred about the centre of mass of the system rather 



than about the nucleus itself: this merely involved replacing 
the mass of the electron in the Bohr formula by the reduced 
mass tl = m m l[m +m }, The second step was taken by 

' b p e p 

Sommerfeld: in the Bohr-Sornmerfeld atom the orbits are 
allowed to be elliptical, and the degree of ellipticity is 
determined by a further quantum number k, the azimutbal 
quantum number: but the energy of the orbits was found to 
be independent of their eccentricity. The third improvement 
was also made by Sommerfeld: he incorporated relativity into 
the model, and found that its effect was to cause a mismatch 
of the ends of the elliptical orbits, so that the electron de- 
scribed an open orbit around the nucleus— a continuously 
evolving orbit that resembled a rosette. The inclusion of 
relativity caused the energies to depend weakly on k, and 
quite remarkable agreement with experiment was obtained 
(the numbers obtained are the same as those obtained in the 
"Dirac theory of the hydrogen atom). 

Although the numbers are almost exact, the model of the 
hydrogen atom from which they are obtained is fundamentally 
wrong, and we are forced to the view that the agreement with 
experiment is an astonishing coincidence: this coincidence 
probably stems from the very peculiar properties of the 
Coulomb potential, properties that remain even in modern 
quantum-mechanical theories of the atom. The fallacy in the 
model was indicated by the later developments of quantum 
theory, for there it is discovered that the concept of trajectory 
is alien to phenomena on an atomic scale (see "uncertainty 
principle). Therefore it is wholly false to attempt to discuss 
the dynamics of a system in terms of the trajectories of its 
components: the Bohr orbits and the Bohr-Sommerfeld orbits 
are macroscopic concepts that have no meaning on the scale of 
the hydrogen atom. Furthermore, it is quite clear that the 
Bohr model is incomplete in the sense that in its postulates it 
virtually asserts the structure of the hydrogen atom and no 
justification is given for the stationarity of states and the 
quantization of angular momentum; these are provided much 
later by the theoretical structure of quantum mechanics. 
Nevertheless, Bohr's achievement was considerable, for it 
applied to a problem in mechanics a theory that had been con- 
structed on the basis of the behaviour of radiation, and was 
therefore one of the first germs of the view that optical and 
mechanical phenomena were essentially identical. 



bond 



25 



Questions, I. State the postulates of the Bohr theory of the 
hydrogen atom. Which of them conflicted with the require- 
ments of classical mechanics? What is the form of the ex- 
pression for the energy that is obtained on this model? 
Deduce an expression for the frequency of the transitions of 
the hydrogen atom: do these conform with the known 
"hydrogen atom spectrum? How was the original mode! re- 
fined? What is the significance of the quantum number k? The 
elliptical orbitals with k = (which are straight lines swinging 
through the nucleus) were rejected as implausible: can you see 
the connexion of, these rejected orbitals with the s-orbitals of 
the modern theory of the atom? The lower energy state of the 
Bohr theory requires the presence of an angular momentum to 
repel the electron from the nucleus: does the quantum- 
mechanical theory lead to the same conclusion? Why is the 
Bohr theory untenable? Discuss the role of the uncertainty 
principle in the Bohr theory. Why was the theory so 
important? 

2. Deduce the energy of the hydrogen-atom stationary states 
on the basis of the Bohr theory. Set up an expression for the 
potential energy of the electron, and then relate the 
centrifugal force to the angular momentum. Balance the two, 
and then replace the angular momentum by nli. Compare your 
result to the accepted expression for the °R yd berg constant. 

Further information. See § 1.2 of Herzberg (1944} and Chapter 
2 of Pauling and Wilson (1935( for information „bout calcu- 
lations on the hydrogen atom. For a view of the hydrogen 
atom in an historical perspective, see Jammer (1966). 

bond. The nature of the chemical bond— the reason why 
atoms stick together and form molecules of a definite shape 
and energy— is one of the central topics and successes of the 
application of quantum theory to chemistry. Elementary 
chemistry distinguishes three kinds of bond between atoms: 
the ionic (where electrons are transferred between atoms and 
the bond is the electrostatic interaction between ions), the 
covaient (where electrons are donated by both partners, and 
shared more or less equally), and the dative (where one 
partner donates both electrons, which then are shared). 
Modern quantum chemistry shows how these three types may 
be considered to be special cases of a general form of bond. 



Elementary chemistry also ascribes the tendency to bond to 
the tendency of atoms to 'complete their octet' or to achieve 
an 'inert gas configuration'. Modern quantum chemistry 
interprets these rules of thumb in terms of the quantum- 
mechanical properties of electrons and nuclei. 

Atoms group together and form molecules if by so doing 
the system attains a lower, more favourable energy; therefore 
we must seek the reason why energy is reduced when a 
molecule is formed. Generally the different stereochemical 
configurations of atoms differ considerably in energy from 
each other, and so the shape of the molecule is normally 
well defined, and corresponds to the stereochemistry that 
enables the system to attain its lowest energy. Conversely, a 
molecule falls apart into its components, or groups of com- 
ponents, if enough energy is injected into the structure (by a 
collision with another molecule or wall, or by the absorption 
of light), so that its total energy exceeds that of its separated 
components. Energy determines everything: to understand 
the shape and stability of a molecule we must study its energy. 
Why energy is important is of course a much deeper problem. 

In order to assess the contributions that lead to a lowering 
of energy we should remember that the energy consists of 
two parts, kinetic energy and potential energy, A careful 
analysis of the contributions these make is very difficult, and 
quite often people ignore the contribution of kinetic energy 
by presuming that it can look after itself, or that the dominant 
contribution to binding energy is the lowering of the potential 
energy that occurs when electrons and nuclei are brought 
close together. Ignoring the kinetic energy is a dangerous 
game; and makes the description of chemical bonding look 
simpler than it really is. Nevertheless, we shall play the game 
because the situation has been fully analysed in only one 
place, to which we return later. 

Given a kit consisting of two protons and one electron, 
where should the electron be put in an attempt to form a 
stable molecule (of Hj)? The conventional argument is that 
simple electrostatics suggests that the electron should be 
placed in the intemuclear region; then the internuclear 
Coulombtc repulsion will be overcome by the attraction 
between the electron and each nucleus, and the H| species 
will achieve stability on this account. Naturally the electron 
does not congregate solely on the midpoint itself, and the 



26 



bond 



structure should be envisaged as a distribution of charge 
around the two nuclei, with a significant accumulation in the 
internuclear region. The addition of a second electron, to 
form H 2 , will lead to a stronger bond if it too enters the same 
region of space, so that the nuclei can now stick to a double 
helping of opposite charge. 

On the basis of the preceding analysis the structure of the 
hydrogen molecule may be envisaged as two nuclei surrounded 
by a charge cloud of two electrons, with an accumulation of 
charge density in the internuclear region, the bonding region. 
The characteristic bond length of H 2 is the point at which an 
equilibrium: is reached between the repulsive interaction of the 
nuclei, which increases as the bond shortens, and the attractive 
interaction with the internuclear electrons; at very short bond 
lengths the electrons cannot accumulate in the bonding region 
and so the repulsion dominates. If a third electron is added to 
an H 2 molecule it attempts to cluster close to the nuclei, but 
cannot penetrate the bonding region because of the presence 
of the original pair of electrons (it is excluded by the 'Pauli 
principle). It therefore congregates as a fuzzy accumulation 
outside the internuclear region. The force this electron 
exerts on the nuclei is disruptive, and so its presence tends 
to lengthen and weaken the bond. A fourth electron succeeds 
in breaking the bond. 

On this description of the covalent bond ', where the bond- 
ing electrons are provided equally by the two atoms, it is clear 
that two electrons give the strongest bond. Forming strong 
bonds is energetically favourable, and therefore atoms tend to 
form as many as they can without drawing on the inner, tightly 
bound electrons. This situation is what should be held in mind 
when one makes the remark that 'atoms share electrons in 
order to complete their octets'. Notice also that the two 
electrons have to occupy the same region of space to be 
effective in bonding, and in order to do so their spins must be 
opposed: this is a consequence of the -Pauli exclusion prin- 
ciple. This feature underlies the importance of the electron 
"pair in chemical bonding. Electrons do not seek to pair for 
some transcendental reason, nor because they lose repulsive 
energy by pairing— in fact it requires energy to push two 
electrons into the same orbital: they pair in order to attain a 
distribution that leads to the lowest energy for the system, and 
at the bottom of a stack of distributions often lies a molecule. 



This description of the role of an electron pair is seen very 
clearly in both the "molecular-orbital and °valence-bond 
theories of molecular structure. 

In a heteronuclear bond (a bond between two different 
atoms) the situation is analogous, but is modified by the 
possibility that the energy will decrease if the bonding pair of 
electrons accumulates closer to one atom than the other: they 
congregate more in the vicinity of the more "electronegative 
atom. This situation may be envisaged in terms of prising off 
one of the valence electrons of the less electronegative atom 
(the atom with the smaller "ionization potential) and shifting it 
towards the atom with the larger 'electron affinity. This 
process leads to a polar bond, and in one language (the valence- 
bond) it would be possible to say that the pure covalent bond 
is contaminated by ionic components. (Alternatively we 
might say that the molecular °wavefunction is a "superposition 
of covalent and ionic wavefunctions.) Atoms with the greatest 
electronegativity tend to be those that differ from a closed- 
shell configuration by only one or two electrons; and so once 
again we can understand that the tendency to form an octet is 
a manifestation of a search for the lowest energy distribution 
of the electrons. 

When the electronegativities of the two atoms of a 
heteronuclear bond are very different, such as when an atom 
of low ionization potential (left-hand side of the periodic 
table) Is next to an atom with a high electron affinity (right- 
hand side of the table) the stability gained by transferring a 
whole electron from one atom to the other may be very large, 
and the juxtaposition of the two atoms leads to the flow of an 
electron from one to the other, so that side by side there is a 
positive ion and a negative ion: these stay stuck together 
simply by a Coulombic interaction between the charges. This 
extreme case of a polar bond is the ionic bond. It is important 
to note that all ionization potentials are greater than all 
"electron affinities (for atoms), and therefore the Coulombic 
attraction between the ions provides the energy for the forma- 
tion of ionic bonds. 

Apart from the polarity of bonds the most significant dif- 
ference between an ionic and a covalent bond is the directional 
properties of the latter in contrast to the lack of directional 
properties of the former. This arises because the Coulombic 
interaction between two charges is isotropic (the same in all 



bond 



27 



directions), so the structures that can be formed, which are 
often extensive aggregates, are governed largely by the steric 
problem of packing together ions of various sizes: the ionic 
bond gives rise to rigid and extensive crystal lattices with 
characteristic packing patterns. In the pure covalent bond the 
interaction is by no means isotropic because its strength 
depends on the ability of an atom to provide electrons in the 
region between itself and its bonding neighbour. For a 
diatomic molecule there is no problem in principle, but as 
soon as three atoms are considered one encounters the reason 
why eovalently bound structures have a geometry determined 
by the electronic structure of the bonding atoms rather than 
the geometrical problem of packing them together. Taking 
oxygen as an example we can understand that the molecule 
HO can be formed by the hydrogen and oxygen atoms each 
donating an electron to form a polar bond, and then a 
second hydrogen atom froms a second bond to yield HOH; 
the lowest energy configuration of this molecule occurs when 
the second bond is at 1 04-5 ° to the first because the oxygen 
atom can form the strongest bonds at that angle (see •hybrid- 
ization). At this point the oxygen octet is complete, and the 
addition of another hydrogen atom leads to an unstable H 3 
molecule {that is H3O possesses more energy than separate 
H 2 + H), All that water can do is form a dative bond, where 
it supplies both electrons, and this it does to form H 3 0* (in 
which the oxygen "lone-pair electrons donate towards a bare 
proton) oraoxro-complexes with ions. The water molecule is 
therefore a well-defined, discrete entity. The valence of an 
atom, the number of bonds that it may form, is moderately 
well defined for most covalent compounds, and the stereo- 
chemistry is determined by the ability of the atom to provide 
electrons to attain this valence. This is very clearly brought out 
in the "molecular-orbital theory of molecular structure and the 
theory of "hybridization. 

All the preceding discussion is based on the conventional 
view in the fourth paragraph; that, it seems reasonably 
certain, is a sweet seduction. There are few cases where the 
molecular structure has been studied in sufficient detail to 
enable the true source of bonding to be analysed critically, but 
with Hi it has been possible to draw disconcerting conclusions. 
These conclusions run counter to most of the simple accounts 
of the chemical bond and, as far as I know, counter to any- 



thing to be found in textbooks. In Hj the source of bonding 
appears to be a subtle interplay between the kinetic and 
potential energies of the electrons. As H*and H are brought 
together the accurate wavef unction shows that electrons are 
indeed shifted into the internuclear region, but that this leads 
to an increase in their potential energy, in contrast to the 
supposition that their potential energy would decrease if they 
could be shared by both nuclei. On reflection, of course, we 
should realize that an increase is more plausible than a de- 
crease, because the lowest potential energy arises when the 
electrons are as close as they can get to one or other of the 
nuclei, 

Where then does the bonding energy come from? First, we 
should note that the situation is slightly relaxed by virtue of 
the greater domain of freedom open to the electron when two 
protons are present, and consequently its kinetic energy due 
to its motion parallel to the bond falls; but only at first, for 
more complication is to come. A larger decrease in energy 
(increase in bonding energy) comes from a contraction which 
occurs in the atomic orbitals on each nucleus: this contraction 
enables the electron to approach the nuclei more closely, and 
so its potential energy falls: this contribution is very large and 
is the dominant change in the potential energy. On contraction 
there is a price to pay, because as the electron is confined to a 
smaller domain its "kinetic energy rises, and this almost can- 
cels the decrease in potential energy; but not quite, and the 
earlier decrease in kinetic energy along the axis helps to 
counterbalance the change. Finally, the form of the wave- 
function shows that there is also a shift of electron density 
from regions outside the nuclei into the bonding region as 
the atomic orbitals are polarized; this reduces the potential 
energy and increases slightly the kinetic energy. The net 
effect is a large decrease in the potential energy, which is 
dominated by the orbital contraction, and a large increase in 
the kinetic energy, also dominated by the orbital contraction, 
which does not quite succeed in winning. The overall effect is 
that H 2 has a lower energy than H + +H, and so is a stable 
species. 

We stress that this complicated story has been elucidated on 
the basis of a careful study. of the H2 wavefunction, and might 
need to be modified for more complex species. But it is an 
excellent example of the power of myth in chemistry, and 



28 



bond order 



shows the importance of detailed and accurate calculations in 
discovering the true nature of the chemical bond. 

Questions. What determines whether atoms will stick 
together and form a molecule? What thermodynamic quantity 
is the measure of the strength of a chemical bond? What 
happens when a large amount of energy is transferred to a 
molecule? How may the energy be transferred? What feature 
of the distribution of the electron in the hydrogen molecule 
accounts for its stability on a simple model? What contribution 
to the total energy does this description ignore? Why is the 
hydrogen molecule more stable than the hydrogen molecule- 
ion H^, but H^less stable than both? Can this argument be 
extended to the explanation of why two helium atoms do not 
form a stable molecule? Indeed, does it apply to the stability 
of bonds between all closed-shell species, for example the rare 
gases? What change in the distribution of the electrons occurs 
when a homonuclear bond is replaced by a heteronuclear bond? 
Is there an additional contribution to the binding energy? 
What determines the extent of polarization of the electrons in 
the tend? When dees an almost [jure ionic bond occur? What 
is the source of the stability of an ionic crystal: why does not 
a crystal of common salt blow apart into a gas of sodium and 
chlorine atoms? Why does the crystal have a definite structure? 
Why does covalency lead to discrete molecules, and ionic 
bonding to extended arrays of atoms? What determines the 
shape of ionic and covaient species? What determines the 
valency of atoms in covalent and ionic compounds? What 
other type of bonding leads to an extended array of atoms 
with a structure determined largely by packing consider- 
ations? Discuss the likely true cause of bonding in H 3 . 

Further information. See MQM Chapter 10 for a resume of 
bonding theory and more details of its quantum theory. For a 
simple account of the structure of molecules see Coulson's 
The shape and structure of molecules {OCS9) and Coulson 
(1961). See also Murrell, Kettle, and Tedder (1965), and for 
an original classic, well worth reading for the way it teaches one 
to think about the application of quantum theory to real 
chemical problems, see Pauling (1960). A close analysis of the 
nature of the reduction of energy when a bond is formed has 
been given by Ruedenberg (1962), and this is extended with a 
careful discussion of the structure of H 2 by Feinberg, 



Ruedenberg, and Mehler (1970). See in particular the analysis 
on p. 54 of this reference. For more information about the way 
that the concepts mentioned here are developed in quantum 
mechanics, and therefore made amenable to quantitative cal- 
culation, see the entries on "molecular-orbital theory, 
°valence-bond theory, "antibonding, -electronegativity, 
'hybridization. One type of bonding ignored in the discussion 
was that responsible for the structure of metals: see "band 
theory of metals for a short account. 

bond order. The bond order is a measure of the single-, 
double-, or triple-bond character of a bond. In °valence-bond 
theory it is determined by calculating the proportion of single, 
double, and triple bonds in the contributing structures. In 
molecular-orbital theory, which now provides the more 
common definition, it is defined in a slightly more subtle 
manner. The basis of the definition (which is enough for a 
qualitative understanding) relies on the cancellation of the 
effects of occupied bonding and "antibonding orbitals. Thus 
in Hj, where one O-bond is fully occupied, the O-bond order 
is unity; in Hei, where the O-bond and its antibonding 
counterpart are both fully occupied, the bond order is zero. 
In ethene a O-bond and a TT-bond are fully occupied, and so 
the overall bond order is 2 (a 'double bond'/, and in oxygen 
one TT-bond is cancelled by a TT-antibond {see "molecular- 
orbital theory for details) and the rump, one O- and one 
TT-bond, gives an order 2. This idea may now be made more 
sophisticated in order to accommodate fractional bond orders. 

In the °LCAO description of molecular orbitals a bond is 
formed when two atomic orbitals on neighbouring atoms 
overlap and interfere constructively. If the coefficient of the 
orbital on atom A isc« and on atom B is Cg, then the con- 
tribution to the bonding will be proportional to the product 
c.Cn. The order of the bond between A and B is then defined 
as the sum of c.Cn overall the occupied orbitals. When c^ and 
c R are simultaneously large for a particular orbital a large con- 
tribution to the total order results (but it cannot exceed unity); 
when c. and eg have opposite signs, corresponding to an 
"antibonding character between A and B, the product is 
negative— it subtracts from the overall result and so reduces 
the bond order. As an example, each C— C bond in "benzene 
has a contribution of = from the TT-orbitals, and 1 from the 



Born-Oppenheimer approximation 



29 



o-orbitals, and so each bond is of order lE. The C— C bond in 
ethane is of order 1, in ethene 2, and in ethyne 3, in accord 
with a norma! chemical appreciation of the order of the bonds. 

One principal application of the quantitative definition of 
bond order is to the estimation of bond lengths, especially of 
C— C bonds, An empirical relation has been found between the 
length R and order 6qq of C— C bonds, and the length satisfies 
the rough rule /? cc = T665- 0-1398 (1 + b QC ), with ff cc 
in angstroms (see Table 2), The correlation should take into 
account the different hybridizations of the atoms involved in 
the bond; the effect of so doing is that for a given bond order 
the bond length decreases by 004 A on going from sp- 3 to sp- 2 
hybrids, and then by a further 004 A on going to sp-hybrids. 
A further application of bond order is to the definition of 
"free valence. 

Questions. What is the valence-bond definition of bond order? 
Calculate the bond order in -benzene on this basis, first on 
the assumption that only Kekule structures contribute, and 
then on the assumption that 20 per cent of the total structure 
is Dewar-like, What is the molecular-orbital definition of 
bond order? What contribution does an antibonding orbital 
make? What is the bond order in H 2 and He 2 ? What is the 
bond order in Hj? On the basis of the molecular-orbital 
coefficients given in Fig. B10 on p. 20 calculate the bond 
order for benzene, benzene", and benzene*. Estimate the 
bond length of naphthalene in which the TT-bond orders are as 
follows: 0-725 (for 1-2), 0-603 (for 2-3), 0-554 {for 1-9), and 
0-518 {for 9-10). 

Further information. See Coulson's The shape and structure of 
molecules {OCS 9) and Coulson (1961). A helpful and lengthy 
discussion is provided in §6.7 of Streitweiser (1961 ). See also 
Coulson (1959) in the Kekule symposium and Pilar (1968). 

Born-Oppenheimer approximation. The Born- 
Oppenheimer approximation assumes that the electronic dis- 
tribution in a molecule can be evaluated in a static nuclear 
framework. The assumption is based on the great differences 
of mass of the electrons and the nuclei: it is assumed that if 
the nuclei move the electrons can adjust their distribution 
instantaneously to take into account the new potential, and 



that the nuclei are insensitive to the rapid fluctuations of the 
electrons in their orbitals. 

The practical effect of the approximation is that it is poss- 
ible to simplify both the discussion and the calculation of 
molecular electronic structures. Instead of having to treat all 
the particles in the molecule on an equal footing, it is possible, 
according to the approximation, to set the nuclei into a frozen 
conformation, and then to calculate the electronic energy and 
distribution corresponding to it. The nuclei can then be moved 
to a new conformation, and the electronic calculation repeated. 
In this way it is possible in principle to calculate the energy for 
all possible arrangements of the nuclei, and then to find the 
one corresponding to the lowest energy— the stable confor- 
mation of the molecule. 

The Born-Oppenheimer approximation makes the molecular 
potential -energy curve a meaningful quantity: as the nuclear 
conformation is changed the molecular energy also changes, 
and the dependence of the energy on the conformation is the 
molecular potential -energy curve. For a diatomic molecule 
the curve is a plot of energy against bond length, and for a 
polyatomic molecule the curve is a complicated potential - 
energy surface. Such a curve corresponds to a potential 
energy because if the molecule is released from a non- 
equilibrium conformation it wilt spring back into equilibrium 
(or at least vibrate around the equilibrium point), and so the 
rise of energy with changing conformation corresponds to the 
acquisition by the molecule of a potential energy. It should be 
clear that this description relies upon the validity of the Born- 
Oppenheimer approximation, for only then are we able to talk 
about the molecular energy as a function of the parameter 
determining the conformation of the molecule (bond length 
for a diatomic). If the Born-Oppenheimer approximation were 
to fail (tf we were dealing with light or rapidly moving nuclei) 
the notion of a potential-energy surface would fail, and so too 
would the idea of bond length and bond angle. In practice, the 
approximation fails slightly, and small spectroscopic conse- 
quences are observed. 

Questions. State the Born-Oppenheimer approximation. Upon 
what is it based? When might the approximation fail? What 
simplification does it introduce? Discuss the concept of a 
molecular potential-energy curve for a molecule. Calculate 



30 



boson 



the relative velocities of an electron and a proton each with a 
kinetic energy of 100 kJ mof '. 

Further information. See MQM Chapter 9 for a brief dis- 
cussion; for an account of the approximation with some 
mathematics see p. 252 of Slater (1963). For spectral conse- 
quences of the failure of the approximation see King (1964). 
The original paper is Born and Oppenheimer (1927). 



boson. A boson Is a particle possessing an intrinsic °spin 
"angular momentum characterized by an integral spin 
quantum number, including zero. Examples include the 
deuteron 2 H {/ = 1 ), the 4 He nucleus, or oc-particle, the 4 He 
atom (/ = 0), and the "photon [/ = 1). Bosons are not re- 
stricted by the "Pauli exclusion principle (in contrast to 
•fermions), and any number may occupy a single quantum 
state. They do obey the "Pauli principle itself, which demands 
that a 'wavefunction be symmetrical under the interchange 
of any pair of identical bosons. Because many bosons may 
occupy a single state, at low temperatures peculiar properties 
arise; these include superfluidity and superconductivity (where 
pairs of electrons, fermions, behave like bosons). The operation 
of "lasers depends on the "photon being a boson, for an 
intense monochromatic beam of light consists of a large 
number of photons in the same state. 

Further information. See "spin and the "Pauli principle for a 
further discussion. The table of nuclear properties on p. 277 
reveals at a glance which nuclei are bosons and which fermions. 
The way bosons occupy states is taken into account by the 
Bose-Einstein statistics which are discussed in Gasser and 
Richards' Entropy and energy levels (OCS 19), and in 
Chapter 6 of Davidson (1962), §9.6 of Reif (1965}, and 
Chapter 22 of Hill (1960). For a discussion of the fundamen- 
tal role of the distinction between fermions and bosons see 
the article by Peierls in Salem and Wigner (1972) and also see 
Pauli (1940). For accounts of superfluidity see Chapter 15 of 
Rice (1967), and for superconductivity see Chapter 1 1 of 
Kittel (1971 ) and Rose-lnnes and Rhoderick (1969). The 
question of whether the fermion 3 He can show superfluid 
characteristics {if two stick together) is discussed by Osheroff, 
Gully, Richardson, and Lee (1972). Peierls (loc. cit.) also 



discusses the evidence for all particles being either bosons or 
fermions. 



bracket notation; bra and ket. The bracket notation, 
which by virtue of the division {bra Id ket) gives its name to 
the entities known as bras and kets, was introduced by Dirac. 
The state of a system whose wavefunction is $ (r) is rep- 
resented by the ket \n), and the conjugate $/*{r) by the bra 
</j|. The integral /dr^{r)t// {r) is implied by the symbol 
(m\n), and the integral fdrip* {r}£lij/ {r) by the bracket 
(m\Q,ln),il being some "operator. This elegant notation shows 
very clearly the connexion of wave mechanics with "matrix 
mechanics, and enables the whole of quantum theory to be 
put on deep structural foundations, for the kets may be 
interpreted as vectors in a special kind of space (Hilbert space). 

Further information. See MQM Chapter 4 for the use of the 
notation. An account has been given in §6 of the book by 
Dirac (1958) and discussed further by Jauch in Chapter 9 of 
Salem and Wigner (1972). The subject is a component of the 
structure known as transformation theory or representation 
theory: see Chapter 4 of Davydov (1965), Roman (1965), 
von Neumann (1955), Katz (1965), Kaempffer (1965), and 
Jauch (1968). 

branch. The "rotational transitions that occur when a mole- 
cule makes a "vibrational transition give rise to a structure in 
the spectrum which can be grouped into branches: when the 
rotational state of the molecule changes from J to J— 1 the 
tines constitute the P-branch; when J is unchanged the lines 
constitute the Q-branch; and when J changes from J to J+\ 
the lines constitute the R-branch. In "Raman spectra the 
vibrational transitions may be accompanied by changes of 
± 2 in the rotational quantum number: the resulting lines 
form the O-brancb and the S-branch (for J — * J— 2 and 
J — >J+2 respectively). The Q-branch of vibration-rotation 
spectra is absent when the molecule lacks a component of 
angular momentum about its symmetry axis: thus almost all 
diatomic molecules show no Q-branch (the exceptions are 
those with a component of orbital electronic angular momen- 
tum about the internuclear axis, such as NO). The appearance 



branch 



31 



IT 



J 

4 










1 

1 

! 




i 


1 ! 

II. 




i 




1 E 1 

i i I 




4 




! 

1 


i ' 1 

i i i 

i i i 
i i 

i 




3 




i i ! 

i i ! 

i i ! 
i i i 
i i i 




2 


i | 

i i 

1 i 




1 
1 1 
















R — 


n 
















1 
i 


i i 

■ i i 

i ] 






Q 


head 





FIG. B12. Formation of P-, Q-, R-, branches; note the head on the 
R-branch which arises when the rotational constant in the upper level is 
smaller than in the lower. 



and source of the P-, Q-, and R-branches are illustrated in 
Fig. B12, and the energies of the transitions are given in Box 3. 

When the upper and lower vibrational states have different 
moments of inertia (which is especially likely when the upper 
vibrational state belongs to an electronically excited molecule). 



the spacing of the branch lines changes with the value of J. In 
a diatomic molecule it is common for the moment of inertia 
of the upper state to exceed that of the lower, and in this case 
the lines of the R-branch converge as J increases, and may even 
pass through a head at high frequency. The reason for this 
behaviour can be traced to the [B 1 — B")J 2 term in Box 3, 
which outweighs (36' — S'V at high J. An R-branch show- 
ing this behaviour is said to degrade to the red. When the 
moment of inertia is smaller in the upper state the P-branch 
shows the head and degrades to the violet. 

The method of combination differences is used to extract 
the "rotational constants of the upper and lower states. 
Choosing the appropriate pairs of lines in the R- and P- 
branches (see Box 3} and plotting their energy differences 
against J" gives B' or B" from the slope of the line. 

Questions. What is meant by a branch in a vibrational 
spectrum? What is the classification of the branches? To what 
rotational transitions do they correspond? When does a mole- 
cule show a Q-branch? Why should the Q-branch consist of a 



Notation: 



BOX 3: Branches 



J 

J" 
B' 
B" 



rotational quantum number of upper states 
rotational quantum number of lower states 
rotational constant of upper state 
rotational constant of lower state 
hv energy of vibrational transition. 

R-branch (high energy; AJ = +1 ;J =J + 1 1 

A£ = hv + 2B' + (38' - B")J" + {$' - B")j" 2 = hP + RU") 
Q-branch (&/ = 0;/ =/) 

t£ = h»+\B'~ B")J' + (ff' - B")/ 2 «*f>V+ Qtf) 
P-branch (low energy; £J — — 1 ; J' = J — 1 ) 

££=h»-[B' + B")J" + (6' - B")f 2 =hv + P{J") 
If B' < B" the R-branch may form a head; if B' > B" the 
P-branch may form the head. 
Combination differences 

mj")-py") = 4B'v" + h 

rv"-})-pu"+\) = 4b"u" + h . 



32 



Brillouin's theorem 



set of very closely spaced lines? On the basis that the energy 
levels of a rotating diatomic molecule are given by the 
expression E{v, J) = (c+ j)ht<J + B v J[J 4- 1), comment on 
why the rotational constant B should be labelled with the 
vibrational quantum number V, and deduce expressions for 
the energy of the transitions of the P-, Q-, and R-branches of a 
rotating molecule. What spectral information can be obtained 
by a study of the branch structure? Now suppose that B 
depends on the electronic state of the molecule. Deduce the 
expressions given In Box 3 for a transition involving electronic, 
vibrational, and rotational excitation, and find the value of J 
for which a head may be formed (assume first /' > /" and then 
/ < / ' , / being the moment of inertia). 

Further information. See MQM Chapter 1 for a discussion of 
rotational structure. For an account of the rotation-vibration 
spectra of molecules, examples of branch structure, and an 
account of the information that they can provide, see Chapter 
4 of Wheatley (1968), Barrow (1962), Whiff en (1972), King 
(1964), and Herzberg (1950). 

Brillouin's theorem. Singly excited states of closed -she II 

molecules do not mix directly with the ground state. (Like 
most statements, this can be made to sound more complicated; 
for those who enjoy such sounds we may state the theorem as 
follows: if a -matrix element of the electron-electron Coulomb 



interaction is calculated between a closed-shell -configuration 
of a molecule and a configuration differing by the excitation 
of a single electron, then that matrix element is zero.) 

The delight of this theorem will be found in the simplifi- 
cation of -configuration interaction improvements to -self- 
consistent field calculations, because it implies that the singly 
excited configurations cannot contaminate closed-shell ground 
states by mixing directly into them. But do not interpret that 
as meaning that there is no mixing at all with singly excited 
configurations, because these may still mix by an indirect 
process involving interaction with an intermediate state. In 
some cases, indeed, it is found that the indirect route is so 
effective that the singly excited configurations are very 
important. 

It should be appreciated that the existence of Brillouin's 
theorem implies the stability of the ground state as calculated 
by self-consistent methods; if ii were false then the ground 
state could be strongly perturbed by close-lying singly excited 
configurations; but as it is true, direct mixing occurs with only 
relatively distant multiply excited levels, and only indirectly 
with the singly excited levels. 

Further information. See §6-3 of Richards and Horsley (1970) 
for a simple introduction and proof of the theorem, and 
Slater (1963) p. 141 and Appendix 4 for a slightly longer dis- 
cussion and proof. See Brillouin (1933). 



c 



character. In chemical applications of "group theory a 
"symmetry operation is generally represented by a "matrix. 
The character \(R) of the operation R is the sum of the 
diagonal elements {trace or spur) of the matrix. (All these 
names are interrelated: Spur is the German for spoor, hence 
trace and character.) Those who know the basic features of 
group theory may wish to be reminded of the following facts: 

(1 ) symmetry operations in the same class possess the same 
character in an irreducible representation of the group; 

(2) the characters of different irreducible representations of 
a group are orthogonal (Note 2 in Box 4); 

(3) the character of an operation is invariant under a 
similarity transformation; 

{4) the character of a reducible representation of a group is 
equal to the sum of the characters of the irreducible represen- 
tations into which it is decomposable {Note 3 in Box 4); 

(5) the characters may be combined into the form of a 
projection operator which when applied to an arbitrary 
function projects out a component that is a basis for an 
irreducible representation of the group {Note 4 in Box 4); 

(6) the character of the identity operation in a particular 
irreducible representation is equal to the dimension of that 
representation; the dimension of the representation is equal to 
the degeneracy of the basis of the representation. 

These properties are summarized in mathematical terms in 
Box 4; further information is given in Further information, 
and a few useful character tables are listed in Table 3 on 
p. 266-267. 

Further information. A sketch of the content of "group 
theory will be found under that heading; details of its 



content, method, and application will be found in MQM 
Chapter 6, and examples of the way it is applied to problems 
of molecular structure and properties in Chapters 9, 10, and 11. 
A list of character tables for alt the common point groups, and 
some of the uncommon ones, with some simple notes on how 
to apply them, has been prepared by Atkins, Child, and Phillips 
(1970). For a thorough analysis see Cotton (1963), Bishop 
(1973), Tinkham (1964), and Bradley andCracknell (1972), 
especially for solids. Books introducing the idea of group 
theory are listed under that heading. 



BOX 4: Character 

1 . Definition Y (/) (R) s tr D (/) (fl) = 2 D U) (fl) 

ll MM 

X 1 ' 1 W is the character of the operation R in the rep- 
resentation T , in which the operation R is represented by 
the matrix d" 1 !/?). 

2. Orthogonality X^ n iR)*^ i) {R) = ftS., 

h is the order (the number of elements) of the group. 

3. Decomposition of representation T 

4. Projection operator for basis f*'* of F 1 " from a general 
function f 

pW/» *<« p ""> = (yMSx""' l*)*fl 

£. is the dimension of I" 1 *''. 

5. Selection rules. The integral JdTf i ' ) *Sl l ^f <k) disappears 
unless reoccurs in the decomposition of V "' X P . 



33 



34 



charge density 



charge density. There are two sorts of charge density: one is 
the density of charge at a particular point In an atom or mole- 
cule, and the other is the charge that may be associated with a 
particular atom in a molecule. The former may be determined 
if the "wavefunction is known, for the probability of finding 
an electron in a small volume element dr surrounding the 
point r is simply li/>(r)PdT. Since an electron carries the charge 
— e it follows that the amount of charge in this region is 
—e\\p{r)\ 2 dT. The charge density (charge per unit volume) at 
the point r is therefore —el i/'fr)! 2 . This charge density may be 
used for a variety of purposes, for example in the evaluation 
of the "Coulomb integral or in the calculation of X-ray 
scattering properties of atoms. 

The other definition lays a much coarser grid on the 
molecule: it does not seek to know all the intimate details of 
the distribution of the electron at each point of space. The 
charge density in a molecule generally means the amount of 
charge (or the density of electrons less the number of nuclear 
charges) on each atom in the molecule. Thus in a homonuctear 
diatomic there is zero charge density on each atom because 
the electrons are equally divided between the atoms, and the 
nuclear charge exactly cancels the electronic charge. In a polar 
molecule electrons may accumulate closer to the more 
'electronegative atom; then the charge density is not uniform. 

Charge density is often calculated from a wavefunction that 
has been written as a "linear combination of atomic orbitals. If 
the amplitude of an orbital in a filled molecular orbital is Ca, so 
that the proportion of that atomic orbital in the molecular 
orbital is \c^\ 2 , then that orbital contributes -elc A I 2 to the 
charge density on atom A if it is occupied by a single electron. 
The total charge density on atom A is calculated by summing 
all such terms for the occupied orbitals on the atom. The 
charge density may be used to calculate the "dipote moment 
of the molecule, and in discussions of its reactivity. 

The analysis of the distribution of an electron in terms of 
its population of atomic orbitals is known as population 
analysis. 

Questions. What are the two definitions of charge density? 
What is meant by population analysis? How may the charge 
density at an atom be calculated if the molecular structure is 
known? The wavefunction for the Is-orbital of atomic 



"hydrogen is i/' ls ('")= (l/JBo^expl— rla ): what is the charge 
density at a point r? What is the total charge density of the 
atom? The wavefunction for the hydrogen molecule ion is 
(1//2) [^ ]s ('" al ) + ^ ls ( r bl )] . where r t is the distance of 
electron 1 from nucleus a and r its distance from nucleus b 

b] 

(note that the molecular orbital is not accurately -normalized: 
does it matter?): what is the charge density on each atom? 
What is the charge density on each atom in the "benzene 
molecule? 

Further information. See "alternant hydrocarbon for a 
further point concerning the Couison-Rushbrooke theorem 
and charge density; see also "dipole moment for the way the 
concept is applied. "Bond order is a related concept of 
population analysis, A discussion of charge density will be 
found in Coulson (1961); Pilar (1968), McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972). Tables of 
molecular-orbital parameters have been prepared by 
Coulson and Streitweiser (1965). See also Streitweiser (1961), 
Salem (1966), Pople and Beveridge (1970), and Murrell and 
Harget (1972). 

Closure approximation. The expressions that appear when 
in order to calculate a molecular property one indulges in 
second-order "perturbation theory normally contain sums over 
all the excited states of the system, and to incorporate them 
properly it is necessary to know the energy of every excited 
state and the wavefunction for each. Normally this is an imposs- 
ible task because neither the excited-state energies nor the 
wavefunctions are known at all accurately. When an exact 
answer is not required, for example, when an order of 
magnitude of a physical quantity is sufficient or when it is 
desired to see in a general way how the property depends on 
various aspects of the molecule, it is possible to make the 
closure approximation. This consists of pretending that all the 
excited states that mix into the ground state have the same 
energy. The propriety of this we shall ignore for the moment; 
but its advantage is that expressions of the form Z/Wof,/V n o/ 
(£ —En ), which reference to Box 1 6 on p. 1 72 shows to be 
central to perturbation theory, are replaced by 
(1/A)i M 0n N no where A is the mean excitation energy. This 
expression may be developed further by recognizing that it 



chemical shift 



35 



has the form of a "matrix product {but see Questions); and so 
the second-order perturbation expression reduces to 
(M/V)oo/A. In order to evaluate this one needs to calculate 
only the ground-state 'expectation value of the product MN, 
and no knowledge at all of the nature or the energies of the 
excited states is required. The step from the sum over /Won^no 
to {MWqo is the 'closure' that gives the approximation its 
name. 

But what price have we paid? The error in the method is 
the absurdity that all the excited states have the same energy. 
When they all lie close together (as in some sense is true in the 
"hydrogen atom, see Fig. G3 on p. 86) the approximation is 
not ludicrous. In other cases the method can be justified 
weakly by saying that the value of A is to be chosen as a 
parameter which relates the true sum to the quantity we can 
evaluate (MN)^; the parameter is then varied in a plausible 
way to simulate the effect of modifications to the molecular 
structure. 

Questions. What is the source of the difficulty of using 
perturbation theory to calculate molecular properties, and 
how does the closure approximation circumvent it? When is 
the closure approximation plausible? The sum over the 
excited states in the perturbation expression is not really a 
true matrix sum because, according to Box 16 on p. 172, the 
ground state has to be omitted; show that the closure 
approximation really leads to an expression with (/WA/)oo ~~ 
MqqNm in the numerator, and so only the ground-state 
properties are required oven with this correction. When M and 
N are identical interpret the numerator in terms of fluctuations 
in the property M (interpret the numerator as a mean-square 
property). Discuss the closure approximation in terms of its 
application to the calculation of electric "polarizabilites, where 
more information will be found. Show, in particular, that the 
polarizability of a molecule should increase with its size. 

Further information. See MQM Chapter 1 for a discussion 
and Chapter 11 for applications of the closure approximation 
to the calculation and discussion of the electric and magnetic 
properties of molecules. Further applications are outlined by 
Davies (1967), who underlines, on his p. 47, the remark made 
by McLachlan (1960) about possible limitations on the appli- 
cability of the closure approximation. 



Chemical shift. In a "nuclear magnetic resonance (n.m.r.) 
experiment an external magnetic field is adjusted until the 
energy required to invert a nucleus (to realign its magnetic 
moment) is equal to the energy of the "photons in the 
electromagnetic field bathing the sample; most efficient 
exchange of energy occurs when the nuclear energy levels are 
in "resonance with the radiation field. The nuclear magnetic 
energy depends on the local magnetic field rather than the 
applied field, and these may differ because the applied field is 
able to induce extra local fields in the vicinity of the nuclei. 
For a given applied field nuclei in different chemical environ- 
ments experience different but characteristic local fields; there- 
fore they come into resonance with the radiation at different 
values of the applied field. Different groups of nuclei therefore 
give rise to absorption lines at different magnetic fields, and 
since the n.m.r. spectrum is a plot of the absorption against 
applied field, different molecules, and even different groups on 



liqhfly shielded 



heavily shielded 




9-806 

low field 

0-?OT 



■f r-scale I 



FIG. C1. The 60 MHz n.m.r. spectrum of acetaldehyde with TMS added 
as reference. The S- and the r-scales for the chemical shift of GHO and 
CH 3 are idicated. Note the fine structure. 



36 



chemical shift 



the same molecule, give rise to absorptions in different regions 
of the spectrum (Fig. CI). The separation of the absorption 
lines is the chemical shift. 

We deal first with the scales on which chemical shifts are 
measured. The separation of the resonances is measured in 
hertz {Hz, cycles per second), but, because the local field 
depends upon the strength of the applied field, to quote a 
separation is insufficient. Therefore the separation is divided 
by the mean frequency, or, what is effectively the same, the 
frequency of the stimulating field v . If one proton resonates 
at a frequency v. and another at v., the value of 5.. = [v.— V.)/v 
is quoted as the chemical shift on the 5-scale. Since the 
separation is generally no larger than about 1 kHz, and the 
spectrometer operating frequency is of the order of 100 MHz, 
the chemical-shift scale extends up to about 1Q~ S . It is con- 
venient therefore to express chemical shifts as parts per million 
{p. p.m.), and so a value of 1 p. p.m. implies that two lines are 
separated by 100 Hz in a spectrometer operating at 100 MHz, 
and by 60 Hz in a spectrometer operating at 60 MHz. The 
absolute chemical shift, the shift of the resonance of a proton 
in a molecule relative to the bare proton, is of little practical 
interest, and a scale often adopted sets the protons in 
tetramethyl siiane (Me 4 Si, TMS) at the origin of the 
chemtcal-shift scale (6 = 0, by definition); heavily shielded 
protons have small 5 values and resonate close to the TMS line; 
lightly shielded protons have large 6 values and resonate well 
to low-field of the TMS line. Another scale is the T-sca/e, in 
which TMS is set at T= 10; it follows that the scales are 
related by r = 10—5. When the rvalue is large the nuclei are 
heavily screened. Some representative rvalues are listed in 
Table 4 on p. 268. 

What are the contributions to the chemical shielding of a 
proton? All the contributions may be ascribed eithento the 
currents induced by the applied field on the atom itself or to 
the currents induced in neighbouring groups. The currents 
induced on the atom may be either paramagnetic or diamag- 
netic (see "magnetic properties), and these respectively 
augment (deshield. move to smaller r) or oppose (shield, move 
to larger r) the applied field. The general explanation of 
'magnetic properties shows that the paramagnetic current is 
small when the excited states of an atom are high in energy, 
and this is the case in the hydrogen atom. For other atoms in 



molecules there are several states that lie quite close to the 
ground state, and in them the paramagnetic effects can domi- 
nate the diamagnetic. Diamagnetic currents are independent of 
the availability of excited states and so always contribute to 
the shielding (but might not dominate it). 

When the currents are induced in a neighbouring group the 
proton experiences an additional field which may be ascribed 
to the induced magnetic dipole moment on the neighbour. 
Thus if the susceptibility of the neighbouring group is Y„ in 
the direction of the applied field, the latter induces a magnetic 
moment X m B which is a source of field at the neighbouring 
proton. Such a dipolar field has a (1— 3cos 2 f?)/ff 3 dependence 
(see Fig. C2), and so the orientation of the proton and the 
induced moment are of crucial importance in determining 
whether the interaction is shielding or deshielding; the sign of 
the shift also depends on the sign of the susceptibility. 
Furthermore, for a tumbling molecule the local-field correc- 
tion disappears unless the magnetic susceptibility of the 
neighbouring group is anisotropic, and so the shift is pro- 
portional to this anisotropy. Two famous cases are illustrated 



-C=C- bond 



shielded 




shielded 



b t 6 

in diced rinij current 




deshielded proton 

FIG. C2. (a) Shows the regions of positive and negative chemical shift 
in the neighbourhood of a double bond. If x,. and \. are the magnetic 
susceptibilities parallel and perpendicular to the bond the shielding 
constant varies as a~(x M - x, I tt-3cos 2 0)/3R 3 . (b) Shows the ring 
current contribution to the chemical shift of ring protons in benzene. 



colour 



37 



in Fig. C2: the position of the resonance in the protons near a 
C=C double bond is determined largely by the magnetic moment 
induced in the TT-electrons of the bond. The effect of the field 
this moment generates is to shield or deshield protons in the 
vicinity, depending on whether they lie respectively inside or 
outside a double cone of half angle 54° 44' (at which angle 
1— 3cos 2 <? = 0). The other example is benzene, where the 
proton shift to low field arises largely from the ability of the 
applied field to generate a ring current when it is perpendicular 
to the plane of the molecule. 

The calculation of the chemical shift normally concentrates 
on the calculation of the shielding constant. This relates the 
local field to the applied field 8 through fi(local) ■ (1—0)6. 
When a is negative the local field exceeds the applied field 
(deshield ing), and the nucleus resonates at low field; when O is 
positive the local field is less than the applied, the nucleus is 
shielded, and the resonance occurs to high field. Just a little 
care is needed to relate a positive or negative shielding constant 
to a diamagnetic or paramagnetic current because the 
neighbouring-group effect depends, as we have seen, on the 
orientation of the molecule, and, depending on the orientation, 
either sign of the shielding constant can arise from either sign 
of the current. 

Questions. 1. What is the meaning of the term 'chemical shift'? 
Is it dependent on the magnitude of the applied field? Is the 
separation of the lines due to protons in different chemical 
environments likely to be smaller or greater in a large applied 
field than in a small field? What happens to the separation 
when the field is changed from 14 kG to 50 kG? Is there any 
advantage in doing n.m.r. spectroscopy at high magnetic 
fields? What is the significance of the statements 'the protons 
resonate at 5 TMS = 2' and 'the protons resonate at2r'? What 
is the significance of a high rvalue? Put the following proton 
resonances in order of increasing field: C$H 6 , CH 3 I, CH 3 OH, 
CfiHu (see Table 4). What is the source of the differences 
between the local field and the applied field? Under what 
circumstances is there a deshield ing effect at the proton? What 
is the role of the neighbouring group of a proton? Why is it 
possible to get both positive and negative chemical shifts from 
both paramagnetic and diamagnetic currents in the neighbour- 
ing group? The magnetic susceptibility of the benzene ring in a 



direction perpendicular to the plane is — 9'5 X 10T 5 and parallel 
to the ring it is —35 X 1CT 5 ; calculate the chemical shift at the 
benzene protons using the formula in the caption of Fig. C2; 
calculate the shift of a proton at a point 3 A above the centre 
of the ring. What is the shielding constant in each case? 
2. Derive an expression for the chemical shift of the protons in 
benzene that takes into account the anisotropy of the mag- 
netic susceptibility of the ring and the free rotation of the 
molecule, and show that the effect of the ring current would 
vanish if the anisotropy were zero. Assume that the field 
induces a magnetic dipole moment at the centre of the ring. 

Further information. See MQM Chapter 1 1 for a detailed 
account of some of the steps that are needed to calculate the 
chemical shift of molecules in terms of the fields and currents 
induced by applied fields. For an account of the role of the 
chemical shift in n.m.r. see McLauchlan's Magnetic resonance 
'OCS 1). Further details will be found in Lynden-Bell and 
Harris (1969), Carrington and McLachlan (1967), Pople, 
Schneider, and Bernstein (1959), Memory (1968), Emsley, 
Feeney, and Sutcliffe (1965), and Abragam (1961). All these 
books deal with the calculation and interpretation of n.m.r. 
spectra in terms of the chemical shift; see also Further 
information in "nuclear magnetic resonance. The calculation 
required in Question 2 is done on p. 59 of Carrington and 
McLachlan (1967). The nature of the ring current is discussed 
by these authors and by Davies (1967). 

colour. A material is coloured if it is able to absorb a band of 
wavelengths from incident white light or if it is able to scatter 
light of one frequency more effectively than another. If we 
discount the blue of the sky the more common mechanism is 
the former. Visible light spans the region from 700 nm (red) 
through yellow (at about 580 nm), green (530 nm), blue 
(470 nm) to violet (420 nm). The energies of the photons vary 
from 1 -7 eV to 3-0 eV over this range (see Table 5), and there- 
fore for a system to absorb in the visible it must possess 
excited states within this distance from the ground state. When 
red is absorbed (by virtue of the presence of low-lying excited 
states) the object appears blue; when blue or violet is absorbed 
it appears red. Low-lying energy levels are not particularly 
common among living systems, and so the predominant 



38 



commutator 



colours of naturally occuring substances tend to be towards 
the red end of the spectrum. This is part of the reason for the 
relative scarcity of purple cows, or bright blue dogs, plants, etc. 
Furthermore, if living systems had low-tying excited states 
they would be uncomfortably liable to photolysis. Chlorophyll, 
however, is specially constructed to cope with this situation, 
and so green is ubiquitous. The most intense colours are due 
to "electric dipole transitions. A chromophore is a group whose 
presence endows colour on a molecule. An important organic 
chromophore is the C—C double bond, in which the relevant 
transition is of an electron from the 7T-bond to the "antibond- 
ing TT*-orbital (a 7T*<-7r, 'pi-to-pi-star', transition). The carbonyl 
group (C=0) is also important: the relevant transition is from 
a nonbonding °lone pair on the oxygen to the 7r*-orbital (an 
'n-to-pi-star', 7T**-n, transition). This transition is electric- 
dipole forbidden {see 'selection rules and "oscillator strength), 
and so is generally weaker than the ir**-ir transition. 

Transition-metal complexes are often coloured: this is a 
consequence of the presence of the d-electrons and their small 
energy splittings arising from the 'crystal field of the surround- 
ing ligands.The intensity of the colour is low because d-d trans- 
itions are forbidden by the -selection rules in operation, and 
the intensity is due to a "vibronic transition. More intense 
colours in the same systems (such as the characteristic intense 
purple of permanganate, or manganate (VII) ion, MnO^) are 
often due to charge-transfer transitions, in which electrons 
migrate from the metal ion to the ligand or vice versa. Such 
transitions possess a large "transition dipole moment. 

Insulators are often colourless and, when crystalline, 
transparent: think of diamond. This is because the electrons 
are tightly bound to the atoms and are available neither for 
conduction nor for light absorption (the conduction "band of 
diamond can be attained only by supplying 5*3 eV of energy). 
As the binding gets less tight, or as the conduction band 
approaches the filled band, colour begins to appear: the band 
gap in CdS 2 is 2-42 eV, and hence it is a yellow-orange solid 
(absorbing in the blue); silicon, with a band gap of only T14 
eV, absorbs all frequencies and has a metallic lustre. 

Metals are characterized by a shining surface when freshly 
cut, and this hardly seems compatible with the remark that 
they absorb all the incident light. The answer lies in the fact 
that they also radiate all the incident light. This can be 



envisaged in terms of the high mobility of the electrons in a 
metal— an oscillating light wave approaches the surface, its 
electric field drives the surface electrons back and forth, and 
the incident light is quenched. But the oscillating surface 
electrons themselves give rise to a radiated light field, and so 
almost all the light is reflected. In the case of some metals 
(copper and gold are familiar examples) there are true absorp- 
tion bands in the visible region, and both these metats extract 
some blue light (and get hotter in the process). 

Two examples of colouring arising through scattering may 
be mentioned briefly. The sky is blue because blue light is 
scattered more strongly than red; therefore more of the sun's 
blue radiation is scattered down to us than is its red (except 
in the late evening, when the sun appears ruddy because 
someone further west is getting its blue light for his daytime 
sky). The reason why clouds appear white even though their 
presence is also seen by a scattering of incident light is to be 
found in the size of the scattering particles (see Further 
information). The other example is the classical colouring of 
glass by the precipitation of colloidal gold: these minute 
particles scatter away the blue component of transm itted 
light, and to the glass there is imparted a rich ruby hue. 

Further information. See Murrel! (1971) for an account of the 
electronic spectra of organic molecules and Chapter 17 of 
Kittel (1971) for a good survey of the optical properties of 
insulators. A most pleasing account of the physical basis of 
colour has been given in a simple article by Weisskopf (1968), 
The photochemical aspects of the absorption of light are 
described by Wayne (1970) and Calvert and Pitts (1966), on 
which is based Table 5. The chemical aspects of light are also 
described by Bowen (1946). The intensity of the absorption 
of light depends, for "electric dipole transitions, on the 
"oscillator strength, and experimentally it is expressed in terms 
of the "extinction coefficient: see MQM Appendix 10.2 and 
Wayne (1970). Concepts, methods, and data concerning 
colour are described in Wyszecki and Stiles (1967). 

Commutator. A commutator of two "operators A and B is 
the difference AB—BA ; it is normally denoted [A,B]- (AB 
means that operation B is performed first, and is followed by 
operation A ', BA implies that A precedes B.) Two operators 
are said to commute if their commutator is zero. The non- 



complementarity 



39 



vanishing of a commutator of two operators indicates that the 
final result of performing two operations depends on the order 
in which the operations are done: operation A followed by 
operation B and B followed by ,4 lead to different results. 
For example, multiplication of a function f(x) by x followed 
by differentiation is different from differentiation followed by 
multiplication: (d/dx)xf(x) ^ x(d/dx)f{x). The commutator of 
(d/dx) and x, [d/dx, x] , is 1 because from the rule for the 
differentiation of a product (d/dx)xflx) is equal to 
fix) + xWdxWW. 

The importance of the commutator in quantum mechanics 
lies in the theory's dependence on "operators: the manipu- 
lations of quantum mechanics must take the possible lack of 
commutation into account; indeed, the very fact that commu- 
tators do not disappear is the feature responsible for the differ- 
ences between quantum and classical mechanics (see "matrix 
mechanics). The "uncertainty principle applies to observables 
whose operators do not commute. 

The commutator of quantum theory is related to the 
Poisson bracket of classical mechanics, and the recognition of 
this connexion is reputed to have been the cause of one of the 
most jubilant moments of Dirac's life. 

The technical importance of the commutator lies in the fact 
that the "eigenfunctions of one operator are also eigenfunc- 
tions of any other operator with which it commutes. 

Questions. What is a commutator? Calculate the commutator 
of (d/dx) and x, of d/dx and d/dy, of d 2 /dx 2 and x 3 , and of 
x(d/dy) — K(d/dx) andz(d/dx) ~-x(d/dz). Demonstrate the 
validity of the following relations: 

[A,B] = -lB,Al 
[A+B.C] = [A,C] + [B,CJ 
[A.IB.C] ] + [B, [C.A] 1 + [C,[A.B1 ] = (Jacobi identity). 

Prove that a necessary and sufficient condition for two 
operators to have simultaneous eigenfunctions (that is, for the 
eigenfunctions of one operator to be eigenfunctions of the 
other) is that they commute. 

Further information. See MQM Chapter 4 for a discussion and 
proof of some of the consequences of the lack of commutation 
of operators. The final Question is answered there (p. 108). 



See the standard texts on quantum theory for a further 
account; for example, Oavydov (1965), Landau and Lifshitz 
(1958a), Messiah (1961), Schiff (1968), and Dirac (1958). For 
deeper accounts see Jauch (1968) and Salem and Wtgner 
{1972). The consequences of noncommutation for exponential 
operators of the form expfiare deveJoped by Wilcox (1967). 

Complementarity. The wave and corpuscular properties of 
'particles' are complementary in the sense that an experiment 
designed to determine the value of a wave-like property 
automatically eliminates the precision with which a 
corpuscular-like property may simultaneously be determined: 
see 'uncertainty principle. Complementarity is the mutual 
exclusiveness of these two types of property: it is impossible 
to demonstrate simultaneously the wave and corpuscular 
attributes of a particle. 

Further information. See "duality, "uncertainty principle, and 
"wave packet for a more detailed discussion. See p. 1 58 of 
Bohm (1951), Kramers (1964), and Jammer (1966). 

Compton effect. Light scattered from electrons shows an 
increase in wavelength which ts independent of its initial wave- 
length but characteristic of the angle through which it is 
deflected. 

On the basis of classical theory it is surprising that only one 
value of the wavelength shift is observed for a particular angle 
of deflection, and the result strongly suggests that a collisional 
process is involved. If it is assumed that a "photon of energy bv 
and momentum A A is in collision with a stationary electron, and 
that both energy and momentum are conserved in the collision 
of the two particles, then it is a simple matter to deduce the 
expression 5X= \falm t)(1— cos 9). SAisthe wavelength shift 
(always an increase), and is the deflection of the light. This 
expression is indeed independent of the initial wavelength, and 
gives a unique value of SXfor a given 6. The agreement shows 
the essential validity of the collisional model, and so it is 
excellent evidence for the quantization of light into "photons, 
and for their behaviour as particles. 

The quantity b/m c is the Compton wavelength A and its 
numerical value is 0024 A, or 2-4 pm (to be precise: 
2'426 309 6 X 10~ 12 m). Therefore even in the backward 



40 



configuration 



scattering direction [0 = 180°} the wavelength shift is only 
4-8 pm, and this smalfness indicates why it is necessary to use 
X-rays or 7-rays, for only then is the shift a significant pro- 
portion of the wavelength: the effect is independent of the 
wavelength, but it is easier to detect. 

Questions, 1. What are the characteristics of Compton scat- 
tering? What features are inconsistent with a classical view of 
the process as an interaction of a charged particle with an 
electromagnetic wave? Why is the effect unimportant at large 
wavelengths? Calculate the Compton wavelength of a proton. 
2. Deduce the Compton formula. Take an initial photon wave- 
length A and a final wavelength X+ 5X; let the electron be at 
rest initially and after the collision have a kinetic energy ^mv 1 
and momentum mv (do everything non-relativistically). Write 
the expression for the conservation of energy during the 
collision (it is an elastic process), and then do the same for the 
linear momentum on the basis that the light is deflected 
through 8, Expand the quantities to first order in SK and 
eliminate v to get the final expression. 

Furtfter information. See §2.8 of Bohm (1951 ) for a discussion 
of the differences between the classical and quantum situations 
and Chapter 4 of Jammer (1966) for an historical perspective. 
The original work is described by Compton (1923). More com- 
plicated treatments of the Compton effect are to be found in 
§22 of Heitler (1954), Chapter 1 1 of Jauch and Rohrlich 
(1955), and Schweber (1961). Note that the non-relativistic 
limit of Compton scattering is normally called Thomson 
scattering, and the Klein-Nishina formula was the result of the 
first calculation of the cross-section for relativistic photon- 
electron scattering. These matters are discussed in the cited 
references. 

configuration. The electronic configuration of an atom or 
molecule is the description of the way the electrons are 
distributed among the available orbitals. Thus the configur- 
ations of the first row atoms are H Is; He Is 2 ; Li Is 2 2s; 
Be 1s 2 2s 2 ;B ts 2 2s z 2p, and so on up to Ne 1s 2 2s 2 2p*. Some- 
times the inner complete shells are abbreviated to K,L,M, etc.. 
Thus the ground configuration of sodium could be written 
eitheras1s 2 2s 2 2p 6 3soras [KL] 3s. In a similar fashion the 
configurations of molecules may be written in terms of the 



occupied orbitals. Thus we have H 2 1sa 2 ;0 2 IsfJ 2 Iso" 2 
2so 2 2so*- 2P0 2 2plf 2p7r* 2 , and the latter is often 
abbreviated by the weak but wise to . . . o^Tr 2 , If the else- 

9 u g 

tronic configuration is known {for example, by applying the 
"aufbau principle) the spectroscopic "terms may be deduced 
and spectra predicted. 

Questions. What is a "configuration' of an atom? What may be 
deduced if the configuration is known? Write down the con- 
figurations of the atoms He, B, C, O, F, Al, Si, CI, Cu, Fe, U, 
and Cf in an economic fashion. Do the same for the molecules 
H 2 , HD, N 2 , f%, Ne 2 , FeO, and "benzene. 

Further information. The standard collection of atomic 
energy levels, which perforce contains lists of configurations, 
is that of Moore (1949 et seq.). Standard works on atomic 
structure are those of Herzberg (1944) and Condon and 
Shortley (1963). Candler (1964) is helpful. See also MQM 
Chapter 8, the synopsis of -atomic spectra, and the "aufbau 
principle. 

configuration interaction. A xonfiguration tells us how 
electrons are distributed among the available -atomic or 
-molecular orbitals, and the simplest description of the struc- 
ture of an atom or molecule consists of a statement of its 
configuration. Thus molecular hydrogen could be described as 
having the structure '\so 2 . The energy of the molecule corre- 
sponding to this configuration could then be quoted as the 
'molecular energy'. Such a description might not be particu- 
larly good, for a true description of the molecule ought to 
allow for the contamination of the configuration by some 
others: we know that a molecular wavefunction is inaccurate 
for a variety of reasons, and so it is certain that the single- 
configuration description of its structure will be inaccurate in 
some fashion. For example, the single-configuration descrip- 
tion of the molecule of hydrogen is deficient in as much as it 
permits both electrons to be localized on the same nucleus to 
too great an extent. 

The situation can be improved by modifying the wave- 
function by permitting configuration interaction (CI): we 
permit the wavefunction of the molecule to be described by a 
mixture (a •superposition) of wavefunctions corresponding to 
different configurations. Admixture of some excited state con- 



conserved property 



41 



figurations distorts the ground-state function, and if the 
mixture lowers the energy of the ground state then the 
■variation principle states that we have an improved descrip- 
tion of the molecule. In the case of H 2 it turns out that a 
major improvement in the energy, and therefore the wave- 
function, can be brought about by mixing in some of the 
configuration tsCT* 2 : when the effect of this admixture is 
analysed {see Questions) it emerges that the effect of allowing 
configuration interaction is to reduce the contribution of 
situations in which both the electrons are at the same nucleus. 
Those who like the language of electron correlation theory 
will realise that CI has achieved a certain amount of charge 
correlation: simple molecular-orbital theory underestimates 
electron correlations, and the admission of CI goes some way 
to repair the defect. 

What configurations can be mixed into the ground state? 
The first requirement is that they have the same symmetry: 
for example, H 2 is improved by the admixture of '2 rather than 
3 £or 1 I1 The second requirement stems from "Brillouin's 
theorem, and is that singly excited states do not interact 
directly with the ground state if proper -self-consistent field 
configurations are being considered. 

The effect of CI may extend beyond improvement to the 
molecular energy, because the fact that excited-state con- 
figurations are mixed into the ground state means that the 
ground state of the molecule possesses some of the character- 
istics of these excited states. Such contamination may 
influence the predictions of the °Hund rules, for instance. 

The simplest way of doing a CI calculation in practice is to 
calculate the orbitals and their energies, and then feed in 
electrons to form various configurations. The ground state 
corresponds to the configuration with the lowest energy. Then 
the actual wavefunction of the molecule is expressed as a 
•linear combination of these configurations {using the sym- 
metry criteria to decide which configurations one should 
bother about), and then the variation principle is used to 
determine the best mixture. A better, but much more com- 
plicated, procedure is the multkonfigurational calculation in 
which the best structure for each configuration is calculated 
separately {rather than by applying the "at/fbau principle to 
a single set of levels), and then the variation principle is 
applied to these optimized components. 



Questions. 1. Why is a single-configuration description of a 
molecular state possibly a poor description of its actual struc- 
ture? In what respect is a single-configuration description of 
the hydrogen molecule a poor object in simple molecular- 
orbital theory? What is configuration interaction, and why 
does it overcome some of the defects of the single- 
configuration method? What effect does it have on the inter- 
pretation of the wavefunction of the ground state of Hj? 
What other influences on molecular properties may CI have? 
Suggest the influence of CI on the stability, dipole moment, 
magnetic susceptibility, and polarizability of molecules. Will 
spectral selection rules be modified by CI? What are the 
criteria for selecting the configurations able to interact with 
the ground state? 

2. The energy of a configuration A ties at Af above the ground 
state configuration of energy E . Suppose that there is an 
interaction between the configurations (that is, there is a 
matrix element V between the configurations). Use the 
variation principle to show that the revised ground-state 
energy with CI is given by the smaller root of a quadratic 
equation (obtained from the "secular equations). Make a rough 
estimate for Af in H 2 and use a value of V = 1 eV to compute 
the modifications to the curves on CI. Is it reasonable to use 
the same value of V at all internuciear distances? What is the 
source of the interaction responsible for the magnitude of V? 

3. Extend the results of Question 2 to compute the wave- 
functions of the two states. Show that the configuration o 2 , 
with u proportional to 1s g + 1s b , is modified by the admixture 
of the configuration a* 2 , withe* proportional to 1s a — 1s b , in 
a way that can be interpreted as an improvement of the dis- 
tribution of the electrons. Discuss the role of CI on the 
products of dissociation of the H 2 molecule. 

Further information. See MQM for a short discussion of mole- 
cular hydrogen in terms of CI, and see also Coulson (1961). A 
simple pragmatic approach to the subject will be found in 
Richards and Horsley (1970), and a lengthier, but helpful, 
account is given by McGlynn, Vanquickenborne, Kinoshita, 
and Carroll (1972); both work through some examples. 

conserved property. A conserved quantity or property is 
one whose value does not change with time. A familiar 



42 



correlation energy 



example is the total energy of a system (First law of thermo- 
dynamics); another is the angular momentum of an electron in 
an atom, or the component of angular momentum about the 
internuclear axis in a diatomic molecule. An example of an 
unconsented quantity is the angular momentum about an axis 
perpendicular to the internuclear axis in a diatomic molecule: 
this can be envisaged as the electron beginning its journey 
about the perpendicular axis, but colliding with a nucleus 
before it has completed its rotation. This collision changes the 
value of the angular momentum about the perpendicular axis, 
and so it is not a conserved quantity. A conserved quantity is 
also called a constant of the motion. 

The definition of a conserved property can be made 
quantitative by defining it as an observable whose corres- 
ponding "operator "commutes with the "hamiltonian of the 
system. For example, the linear momentum of a system is 
conserved if the linear momentum operator commutes with 
the hamiltonian for the system. In the Questions you are asked 
to show that the commutator of the linear momentum and 
the hamiltonian is proportional to the gradient of the potential 
energy of the system. From classical mechanics we know that 
the gradient of the potential is a force, and so we arrive at the 
pleasing conclusion that the linear momentum is a conserved 
quantity only in the absence of a force. Alternatively, we may 
conclude that in the presence of a force the linear momentum 
is not conserved, or that the linear momentum changes. This is 
essentially the content of Newton's second law of motion. 

Questions. What is meant by the term 'conserved property'? 
Give some examples other than those mentioned in the text. 
Consider the "expectation value of the 'operator corres- 
ponding to some observable £2. By invoking the time- 
dependent form of the "Schmdinger equation, show that the 
rate of change of the expectation value, (d/df) {£2) , is equal 
to (i/li) <[A/,J2] >. This shows that if the commutator of H and 
£1 disappears, then the expectation value of £2 remains 
constant in time. Now let £2 be the linear momentum in the 
^-direction and let H be the operator composed of the 
"kinetic energy plus some potential energy V(x). Deduce an 
expression for the rate of change of the expectation value of 
the linear momentum in terms of the expectation value of a 
force. Recognize and, if your spirit is of that kind, be thrilled 



by the fact that you have deduced Newton's second law of 
motion. Reflect on it by considering the fact that, in the light 
of this result, Newton's law may be regarded as an equation for 
average values, and so realize that classical mechanics is a 
treatment of averages and so ignores the finer details of 
motion. What are the conditions that must be satisfied in 
order that energy, angular momentum, and linear momentum 
bo conserved in mechanical systems? 

Further information. See MQM Chapters 4 and 6 for a short 
discussion of conservation laws. More detailed information 
will be found in Chapter 5 of Roman (1965), and Feynman, 
Leighton, and Sands (1963). On p. 195 MQM discusses the 
fact that the conservation of energy, momentum, and angular 
momentum are features of the symmetry of space. 

correlation energy. If it is well done, a "self-consistent 
field calculation of an atomic or molecular structure in the 
Hartree-Fock scheme generally gives a good answer; but even 
if the calculation is done exactly within the scheme the cal- 
culated energy differs from the true energy. A part of the 
discrepancy lies in the neglect of relativistic effects, which 
might be very large for inner-shell electrons possessing a high 
kinetic energy; but even when this is allowed for there remains 
a discrepancy. The magnitude of this residual difference is the 
correlation energy, and its presence reflects the approximations 
inherent to the Hartree-Fock scheme. A basic approximation 
of the Hartree-Fock scheme is the neglect of the local distor- 
tion of the distribution of electrons, and the averaging of its 
effect over the whole orbital: instead of an electron's orbital 
being distorted in the vicinity of another electron, the whole 
orbital is modified in an averaged way (Fig. C3). Therefore 
the scheme neglects local electron-electron effects— it neglects 
electron correlations. 

The effect of the neglect of correlations is to cause the 
calculated molecular potential-energy curve (see -Born- 
Oppenheimer approximation) to lie above the true curve. 
Nevertheless, the true shape is approximately reproduced, 
except that it is too narrow in the region around the minimum 
(the equilibrium bond length). The minimum of the Hartree- 
Fock curve occurs at about the same position of the true 
curve. Thus the equilibrium geometries are quite well predicted 



correspondence principle 



43 



--firsf qu«s 




first <fieu 




Hcirrree-fbck 
corrected function 



FIG. C3. (a) The actual distortion and (bl the Hartree-Fock pretence. 



if correlation effects are neglected, but the "force-constants, 
and hence the molecular vibration frequencies, are exagger- 
ated. 

Questions. Even the best Hartree-Fock calculation does not 
give the exact electronic energy of an atom or molecule: what 
are the reasons? What is the correlation energy, and why is it 
so called? How must the Hartree-Fock scheme be improved in 
order to regain more accurate results? If a molecular potential- 
energy curve is calculated on the basis of the Hartree-Fock 
scheme, in what way should it be expected that the equilib- 
rium bond lengths and vibration frequencies will differ from 
their actual values? Would you expect the neglect of corre- 
lation effects to be more or less important as the bond between 
two atoms lengthens? What features would you think of incor- 
porating into a wavefunction of the helium atom in order to 
reflect the tendency of the electrons to remain apart? 

Further information. A simple account of the methods that 
are employed to deal with the calculation of atomic and mole- 
cular structures, and the role of correlation effects, is con- 
tained in Richards and Horsley {1970} and in McGlynn, 
Vanquickenbome, Kinoshita, and Carroll (1972). For some of 
the ways the problem has been tackled, see Pauncz (1969), 
Berry (1966), Sinanoglu (1961 ), Clement! (1965), and 
Sinanoglu and Brueckner (1970). 



correspondence principle. At large "quantum numbers the 
mean motion of a system becomes identical with its motion 
calculated on the basis of classical mechanics. 

The principle implies that the rules of quantum mechanics 
contain the structure of classical mechanics when the fine 
details of the situation are ignored. An example of its appli- 
cation is provided by the construction of a 'wave packet to 
represent the motion of a free particle when the energy or 
momentum is only coarsely specified: the packet moves 
along the same trajectory that the same mass point would have 
in classical mechanics. But as the energy becomes more 
precisely specified, so that fewer quantum states contribute to 
its representation, the distribution becomes less classical and 
more quantal. Another example is provided by the Planck 
distribution of energy in a "black-body radiator: as Planck's 
constant dwindles to zero (in a hypothetical classical world) 
the energy distribution becomes that of a classical system and 
agrees with the R ay leigh -Jeans distribution law. Likewise, the 
momentum of a "photon (a light quantum) is transmitted to 
the object that absorbs or reflects it; and when a sufficiently 
large number of photons is involved this impulse is interpreted 
as the steady radiation pressure of classical electromagnetic 
theory. 

The correspondence principle was of profound importance 
in the early days of quantum theory for it acted as a guide 
through the exciting gloom of those days: any calculation 
based on quantum theory had to correspond to a classical 
result in all details when sufficiently large quantum numbers 
were involved and quantum fluctuations ignored. As an 
example of this kind of development we may consider the 
radiation emitted by an harmonically oscillating electron: the 
existence of an array of equally-spaced quantized energy levels 
in a "harmonic oscillator suggests that a very wide range of 
frequencies could be emitted because of the indefinitely large 
number of different energy separations that may be obtained. 
Nevertheless, a classical oscillator, to which the quantum 
harmonic oscillator must correspond, emits only a single 
frequency, that of its natural classical motion. In order to 
reproduce this result it is necessary to impose restrictions at 
the quantum level on the transitions that can occur; thus the 
correspondence principle leads to the "selection rule that an 
oscillator may make a transition only to a neighbouring level. 



44 



Coriolis interaction 



Such rules emerge naturally from the later quantum mechanics, 
but even there the correspondence limit is often a very good 
check on the validity of a calculation. Finally it should be 
noted that purely quantum phenomena disappear in the 
correpondence limit; in particular, all effects due to °spin are 
eliminated. 

Further information. Various classical limits of quantum 
situations are discussed in MQM Chapter 3. For an historical 
perspective and an account of the way the principle was used to 
disentangle the old and discover the new quantum theory see 
Jammer (1966). Applications of the principle are described in 
Kemble (19581 and Kramers (1964). 

Coriolis interaction. The Coriolis interaction is the inter- 
action between the rotation and vibration of a molecule. 
Think first of a diatomic molecule rotating about an axis 
perpendicular to its internuclear axis, if we regard it as a 
classical problem we may imagine the bond lengthening and 
shortening as it vibrates, and this vibration changes the 
moment of inertia of the molecule. 'Angular momentum is 
conserved for the molecule, and so for the same angular 
momentum, but smaller moment of inertia /, in order to 
preserve the product lio it is necessary for the angular velocity 
toto increase. Conversely, as the bond lengthens and the 
moment of inertia increases, the angular velocity must fall in 
order to conserve the angular momentum. Therefore the 
rotation accelerates and decelerates as the bond vibrates. (This 
is the same mechanism whereby managing directors on ro- 
tating chairs, and ice skaters, speed or slow their rotational 
motion.) This picture shows that there is an interdependence 
between the rotation and the vibration of a molecule, and it 
may appear in modifications of the spectral lines. 

An important application of the Coriolis interaction is 
encountered in the case of a vibrating, rotating, linear, tri- 
atomic molecule. There are four "normal modes of vibration 
of this molecule; one is a symmetric stretch in which both 
A— B bonds vibrate in phase, another (named l> 2 ) is the 
asymmetric stretch in which as one A— B bond shortens the 
other lengthens, and the other two are the bending motions 
which may occur in two perpendicular planes {see 'normal 
modes for pictures). Consider the effect of the v 2 vibration 




FIG. C4. The asymmetric stretch (labelled v 2 1 and the molecular 
rotation interact and induce a bending vibration in the plane o! 
rotation. 



interacting with the rotation by the Coriolis mechanism 
(Fig. C4). As one A— B bond shrinks there is a tendency for 
that half of the molecule to speed up; therefore that bond 
tends to bend forward relative to the rest of the molecule. 
Meanwhile the other bond is getting longer, and the Coriolis 
interaction requires that half of the molecule to decelerate; 
therefore it tends to lag behind in the rotation of the molecule. 
The net effect is that the molecule tends to bend in the plane 
of the rotation. But having got this far the stretching motion is 
at the end of its swing, and it begins to swing back: the long 
bond shortens and the short lengthens. The Coriolis interaction 
comes into operation and the faster rotating bond becomes the 
laggard and vice versa. This induces the molecule to bend the 
other way. The net effect of the continuing process is that 
the bending vibration is stimulated by the combined anti- 
symmetric stretch and the rotation of the molecule. This 
means that the two vibrations are not independent and that 



Coulomb integral 



45 



the effect of the Coriolis interaction has been to mix different 
vibrations together. This appears in the spectrum as £-type 
■doubling. 

Questions. What is the Coriolis interaction? Why is it able to 
cause an interaction between the vibrational and rotational 
motion of a molecule? Discuss the changes brought about by 
the change in the moment of inertia of a molecule by virtue of 
the letter's vibration on the angular speed, frequency, 
momentum, and rotational energy of the molecule. Discuss the 
effect of the Coriolis interaction on the vibrations of a linear 
triatomic molecule. What is it unable to mix? What is the 
consequence of the mixing on the spectrum? Apply the same 
reasoning to a planar triangular molecule of the form AB 3 . 
Which vibrations of this molecule is the rotation able to mix? 
Consider the effect of rotations about the axis perpendicular 
to the plane, and about an axis in the plane of the molecule. 
What other manifestations of the Coriolis force can be ob- 
served in macroscopic phenomena? 

Further information. See MQM Chapter 10. The classical basis 
of the Coriolis interaction, which is concerned with the way 
mechanical systems behave in rotating axis systems [rotating 
frames) will be found in §4.9 of Goldstein {1950). Appli- 
cations to spectroscopy will be found in King (1964), 
Herzberg {1945, 1966), Sugden and Kenney (1965), Wilson, 
Decius, and Cross {1955), and Townes and Schawlow (1955). 
A helpful theoretical discussion is to be found in Allen and 
Cross (1963). 

Coulomb integral. The Coulomb integral is the contribution 
of the classical electrostatic interaction between charge dis- 
tributions to the total energy of an atom or molecule; although 
there are some frills which must be added to this basic de- 
scription. 

In atoms the frills are fewer and we deal with them first. 
Consider an electron distributed in an "atomic orbital \p : 

a 

knowing the mathematical form of the orbitals we are able to 
say that in the small volume element dr, surrounding the 
point r, the probability of finding the electron is \\p {r^PdTi ; 
therefore the charge in that region isdgi = -el^ a (ri)rdri. 
Another electron present in the atom may occupy the orbital 
t// ; by the same argument the charge in some little volume 



element dT 2 surrounding the point r 2 is dg^ = — e\\j/ .{rjlPd^. 
If the charges 6qi and dg 2 are separated by a distance r 12 the 
potential energy of their interaction comes from the expression 
for the Coulomb potential, namely dq l dq 2 /$'n€f ) ri2. It follows 
that the two charges at t x and r 2 will give a contribution of 
this form to the total energy of the atom. The total contri- 
bution to the energy can be obtained by summing over all the 
volume elements dT[ and drj, or, since these are infinitesimal 
volume elements, by integrating over them. The procedure leads 
to the Coulomb integral J as illustrated in Fig. C5a. As the 
charges are the same, J is positive, and so leads to an increase 
in the energy of the atom. This means, of course, that the 
Coulomb interaction is repulsive. The frills to tack on to this 
description are concerned with the intrinsic tendency of 
electrons to correlate their motion: the correction to the 
Coulomb energy arising from this effect is termed the 
"exchange energy. 




J =^T-,d Tj \% Cr^nvJ %('$ 



nner shells 



FIG. C5, The contributions to the Coulomb integral (a) in an atom, 
lb) in the VB theory of a diatomic molecule, and Ic) in MO theory of 
(he same. Arrows like "*~*- represent attractive interaction, and - * - *■ 
repulsive. 



46 



crystal-field theory 



In molecules the situation is complicated by the variety of 
interactions possible when several nuclei are present. It is 
further complicated by the existence of two different de- 
scriptions of the chemical bond: the °molecu!ar-orbital and 
the -valence-bond theories. Nevertheless, in each one the term 
'Coulomb integral' signifies the type of energy that arises by 
virtue of the classical electrostatic interaction between charges. 

In the valence-bond method the Coulomb integral is 
composed of three parts: the electrostatic repulsion between 
the two nuclei in the bond, the repulsion between the charge 
distributions on the two nuclei, and the attraction between the 
charge distribution on one nucleus and the opposite charge of 
the other nucleus {see Fig. C5b). This integral is also called J, 
but we note that it may be negative if the last of the three 
contributions exceeds the others. It too must be corrected to 
allow for electron exchange: details are given under "valence- 
bond. 

In the molecular-orbital method the Coulomb integral, 
which especially in the "Hu'ckel method is often denoted a, 
generally consists of four parts. The first is the energy of the 
electron as if it were in an orbital of the isolated atom (this 
therefore contains the attraction of the electron to its own 
nucleus, its repulsion from the other electrons on the atoms, 
and its kinetic energy); the second is the nucleus-nucleus 
repulsion in the bond; the third is the electrostatic attraction 
between the charge distribution and the neighbouring nucleus; 
and the fourth is the repulsion of the electrons on the two 
nuclei. These interactions are illustrated in Fig. C5c and dis- 
cussed further in the entry on "molecular-orbital theory. It too 
must be corrected for "exchange interactions. 

The feature common to all these Coulomb integrals is the 
way they select, from all the interactions within the molecule, 
the ones that would be selected on a simple classical electro- 
static picture of the interactions; they ignore the way that the 
'spin of the electrons correlates their motion. 

Questions. What is meant by the expression 'Coulomb 
integral'? How does the Coulomb integral appear in the theory 
of atoms: present a simple argument on the basis of the 
expected electrostatic interactions, and demonstrate how the 
expression emerges from the proper "hamiltonian for the 
energy of a many-electron atom. Express the energy of the 



helium atom in terms of the Coulomb integral J. Can J (for 
atoms} ever be negative? What additional complications are 
there in molecules? What are the components of the 
electrostatic energy in the valence-bond description of a 
chemical bond between two atoms? What are the correspond- 
ing contributions in the molecular-orbital description of the 
same bond? What features are common to both? What 
features are unique to each? May the molecular Coulomb 
integrals be negative? What would that signify? What additional 
contributions to the Coulomb integral might be expected when 
a molecule containing more than two nuclei is being con- 
sidered? What is the maximum number of different atomic 
orbitals that need be considered inside a Coulomb integral? 

Further information. See MQM Chapters 8 and 9 for the role 
of the Coulomb integral in atomic- and molecular-structure 
calculations. See also Coulson (1961). A helpful account of 
the calculation of Coulomb integrals will be found in McGlynn, 
Vanquickenborne, Ktnoshita, and Carroll (1972). Applications 
to the determination of molecular structure will be found 
there and also in Richards and Horsley (1970). See the entries 
on "exchange energy, "spin correlation, "self-consistent fields, 
and the "molecular-orbital and "valence-bond theories. 

crystal -field theory. The five d-orbitals of a free transition- 
metal ion are "degenerate (have the same energy), but when 
the ion is complexed the ligands remove the spherical 
symmetry of the atom and replace it by a field of lower 




O " 







, 




■6Dq / 

1 / 






4-4 


IfjDq 


4Dq \ 

1 V 




■ 



r 2q Wxy-V-'W 



FIG, C6. The effect of an octahedral crystal field on the energy ol an 
electron in the d-orbitals of the central atom. 



crystal-field theory 



47 




low spin 



-M^" 



1 

strontj fiefd 



weak fieH 



to field 



FIG. C7. The competition between spin-pairing and 100<j leading to 
low-spin or high-spin complexes, 

symmetry. The immediate environment of the ton may be 
octahedral, tetrabedral, and so on. The ligands may be 
regarded as a source of electrical potential, the crystal field, 
and the d-orbital energies adjust accordingly. In an octahedral 
complex the d 2 and d 2 2 orbitals point directly towards 
the ligands (Fig. C6), whereas the other three d-orbitals point 
in mutally equivalent directions between them. It follows that 
the former pair possess one energy and the latter trio a differ- 
ent energy, and so the degeneracy of the orbitals has been 
removed. The energy separation is conventionally called \0Dq. 

Electrons may be fed into the array of atomic orbitals so 
formed, and the structure, spectra, stability, and magnetic 
properties of the complex discussed on the basis of the 'aufbau 
process. The competition that governs the configuration of an 



ion in a complex is between the tendency of the electrons to 
enter the lowest-energy orbitals, which more than three can 
do only if their spins pair, and the tendency for electrons to 
enter a set of orbitals with parallel spins ("Hund rules). Which 
tendency wins is governed largely by the separation of the two 
sets of orbitals, and therefore by the strength of the crystal 
field. A high-spin complex results when the tendency not to 
pair wins, and a low-spin complex results when the electrons 
achieve lowest energy by pairing and entering the low-lying 
orbitals (Fig. C7). 

The crystal-field theory is an approximation because it 
supposes that the energy of ligand-metal bonding is due 
solely to electrostatic effects, and ignores both the covalent 
nature of the bonding and the role that TT-bonding might be 
expected to play: these deficiencies are repaired in the 
broader "ligand-field theory. Nevertheless, the crystal-field 
theory is successful in so far as it expresses the importance of 
the symmetry of the complex in determining the electric, 
magnetic, and chemical properties, and provides a simple rule 
for predictions of these properties. 

Further information. See MQM Chapter 9 for a more detailed 
account. An introduction to the idea of the crystal field is 
provided by Earnshaw and Harrington in The chemistry of the 
transition elements (OCS 13). A pleasing introduction to 
crystal-field theory is provided by Orgel (1960), developed by 
Ballhausen (1962), and consummated by Griffith (1964). See 
-ligand field for further directions. 



D 



de Broglie relation. According to the de Broglie relation a 
particle travelling with a linear momentum p has associated 
with it a wavelength A ■ hip. As the particle's momentum 
increases the associated wavelength decreases. The relation can 
be understood in terms of the interpretation of the 'wave- 
function of the particle, and the connexion between the 
function's curvature and the particle's "kinetic energy. As the 
kinetic energy of the particle increases, the curvature of its 
wavefunction increases; but increasing curvature implies that 
the wave swings from positive to negative amplitude more 
often in a given length: its wavelength decreases. But the 
kinetic energy is proportional to the square of the momentum, 
and so the inverse relation between momentum and wave- 
length emerges. 

A more direct demonstration of the quantum-mechanical 
basis of the de Broglie relation is to observe that the wave- 
function for a particle with -momentum p = ki\ is exp{\kx). If 
this is written as CQ${2~npx/h)+\s'm(2Tipxlh) it can be seen 
immediately that the function corresponds to a superposition 
of real waves each with wavelength hip. 

Questions. What is the de Broglie relation? How can it be 
justified quantum mechanically? Why does the wavelength of a 
particle decrease as its momentum increases? How could the 
existence of the de Broglie wave be demonstrated experimen- 
tally? Calculate the de Broglie wavelength for the following 
particles: an electron accelerated from rest by a potential of 
10 V, 1 kV, 100 kV; a proton travelling at 1 km s _1 , 
1000 km s~';a mass of 1 g travelling at 100 km h" 1 ; a car of 
mass 1 500 kg travelling at 50 km h - ' . Which of these species 



would be expected to show pronounced quantum-mechanical 
behaviour? 

Further information. See Feynman, Leighton, and Sands 
(1963} and Chapter 3 of Bohm (1951). For an account of the 
de Broglie relation by two students of de Broglie, and an 
account of the way that some people are attempting to inter- 
pret it, see Andrade e Silva and Lochak (1969). A pleasing 
account of de Broglie's contribution to quantum theory is 
given in §5.3 of Jammer (1966). 

degeneracy. When two or more different "wavefunctions of a 
system correspond to the same energy they are said to be 
degenerate. Thus the three /7p-orbitals of any free atom con- 
stitute a triply degenerate set of functions, and the TT-orbitals 
of diatomic molecules are doubly degenerate. When only one 
wavefunction corresponds to a particular energy the state is 
said to be non-degenerate ('singly degenerate' is sometimes 
used in this case, but this seems to be an illicit extension). 
'Razing the degeneracy' of a state means removing the 
degeneracy (for example, by applying an electric field to an 
atom). 

The degeneracy of a system is related intimately to its 
symmetry, and whenever a wavefunction can be changed into 
another by a "symmetry operation the functions are degener- 
ate. Thus rotating a free atom through 90° about z is a 
symmetry operation, and as it rotates an np -orbital into an 
np -orbital these are degenerate. In a similar fashion np^ 
may be generated by rotation about another axis, and so it too 

is degenerate with np (and np ). A 3p-orbital cannot be 

k y 



48 



density matrix 



49 



generated by rotating a 2p-orbital, and so, according to the 
symmetry definition, they are not degenerate in atoms— and 
this accords with common sense. 

When two functions correspond to the same energy by a 
numerical coincidence and not by virtue of the existence of a 
symmetry operation they are said to be accidentally degener- 
ate. As an example, it is possible that the 2sO"* and the 2pJT 
orbitals of some diatomic molecule have the same energy to a 
few significant figures; they are then accidentally degenerate. 
A widely quoted example of accidental degeneracy is that of 
the "hydrogen atom, in which all the orbitals of a given 
principal quantum number (that is, thews, Dp, nd, , . , orbitals) 
have the same energy; but this is a fallacious example because 
it is possible to find a cunningly hidden symmetry operation 
by which it is possible to rotate an /js-orbital into an np-orbital, 
etc., and so the degeneracies are true rather than accidental. 
The same 'hidden symmetries' can account for the apparently 
accidental degeneracies of a "particle in a box whose sides are 
in rational proportion (see Question 2), and all exact 'acciden- 
tal' degeneracies can be explained in terms of a deeper scrutiny 
of the symmetry of the system. 

Questions. 1 . When does a degenerate state occur? How is 
degeneracy related to the symmetry of a system? What is 
accidental degeneracy? How many-fold degenerate are the 
ground states of the sodium atom and the boron atom {neglect 
spin-orbit coupling)? What is the degeneracy of the "hydrogen 
atom when the electron occupies a state with principal 
quantum number 1, 2, n? What is the degeneracy of the spin 
state of an electron in zero magnetic field, and what is its 
degeneracy when the field is applied? What can be said about 
the degeneracy of an electron in the d-shell of a transition- 
metal ion when an octahedral ligand field is present? Discuss 
the last problem by symmetry arguments. 
2. Take the energy levels for a "particle in a two-dimensional 
rectangular well (Box 15 on p. 166) and consider the case 
where the sides are equal. Show that the lowest energy state is 
non-degenerate, and that the next- highest state is doubly 
degenerate. Show that the two wavefunctions for the first 
excited state can be interrelated by a rotation of the square 
through a right-angle. Can any of the states be triply degener- 
ate? Can any state of a cubic box be triply degenerate? Now 



let the well have sides a and ca. where c is a rational number: 
find some degenerate states. Are these truly degenerate or 
accidentally degenerate? Is there a hidden symmetry? What 
happens when c is irrational? 

Further information. The best way of discussing degenerate 
situations is in terms of "group theory: see MQM Chapter 5, 
Cotton (1963), Bishop (1973), and Mcintosh (1971). For a 
discussion of the hydrogen atom degeneracy see MQM 
Chapter 3, Englefield (1972), Mcintosh (1971), and Bander 
and Itzykson (1966). The razing of degeneracy is the basis of 
the crystal-field description of transition-metal ions; therefore 
read Earnshaw and Harrington's The chemistry of the 
transition elements {OCS 13), Orgel (1960), Ballhausen 
(1962), and Griffith (1964). Accidental degeneracy is dis- 
cussed in detail by Mcintosh (1958, 1971). 

density matrix. In elementary applications of quantum 
mechanics calculations are based on the "wavefunction \jj; but 
whenever an actual observable is calculated one encounters 
formulae which involve the wavefunction as its square \j/*\p 
or in a bilinear combination of the form ]b*j!/ ; see, for 
example, "expectation value or "transition probability. Why 
not set up a formulation of quantum mechanics that deals with 
the bilinear combinations \b*}jj directly rather than introduc- 
ing them only when an observable is being calculated? Further- 
more, the absolute phase of the wavefunction is immaterial, 
for if both i^ m and $ are multiplied by the arbitrary phase 
factor expi0 the product i^* 1^ remains unchanged, and so all 
observables are independent of the absolute phase; it appears 
that in dealing with the wavefunction one is carrying around a 
piece of useless information. The final preliminary point to 
make is that the state of a system is only very rarely pure: 
normally an actual system has to be treated as a statistically 
large collection of sub-systems, and the result of an experiment 
is determined by some average (actually one of the ensemble 
averages of statistical mechanics) of the products 4l* 4> ; this 

* — i — 

statistical average we denote \p m \p n , and interpret the averaged 
product as the n, m element of a matrix, the density matrix p. 
The "matrix of elements p , which are by definition &*\}j , 
should contain all the information about the system: it carries 
all the pertinent information of the wavefunction, and all the 



50 



dipole moment 



information about the role of the statistical averaging. It is 
not difficult to find an equation of motion of the density 
matrix (corresponding to the "Schrodinger equation for the 
wavefunction), but it is often quite difficult to solve. Never- 
theless, powerful methods exist for dealing with the density 
matrix, and it is a common way of performing calculations 
dealing with the time-evolution of complex systems. 

Further information. For a gentle introduction, see Chapter 
5.4 of Slichter (1963). Then see Chapter 4 of Ziman (1969), 
Chapter 1.8 of Roman (1965), and Fano (1957). For 
applications to molecular and atomic structure, see McWeeny 
and Sutcliffe (1969). 



dipole moment. A positive (+t?) and a negative (— q) 
charge separated by a distance R constitutes an electric dipole 
moment fi = qR. On a molecular scale a separation of the 
order of 1 A (0-1 nm) is a typical magnitude, and the charge 
of the electron a typical charge (4-80 X 10~ 10 e.s.u., 
1-60 X 10" l9 C); therefore molecular dipole moments should be 
expected to have a magnitude of approximately 10" 18 e.s.u. cm 
(3*3 X 10" 30 C m). Dipoles do indeed have this size in many 
systems, and so the unit debye (1 D = 10" 18 e.s.u. cm, 
3-3 X 1Q~ 30 C m) is frequently used when magnitudes are 
quoted. Some typical molecular dipole moments are listed in 
Table 6. A dipole moment is a directed quantity because at 
one end is a positive charge and at the other a negative; in 
order to denote the direction of the dipole in a molecule the 
convention is employed of representing it by an arrow with 
the head at the negative end: H — ► — . Expressions for the 
energy of a dipole when it is in an electric field, and related 
information, are recorded in Box 5. 

The vectorial nature of electric dipole moments is taken 
into account when the overall moment of a substituted mole- 
cule is considered: with each group can be associated a 
directed dipole moment of a magnitude which to a good ap- 
proximation is reasonably independent of the nature of the 
rest of the molecule. The resultant of the vectorial addition of 
the dipole moments for all such groups in the molecule yields 
the overall dipole moment. In this way it is easy to appreciate 
that the dipole moments of chlorobenzene and the o-, m-, and 



BOX 5: Dipole-moment formulae 

Energy of a dipole ju in an electric field E 

£ = — (U.E = — p£ cos 8. 
Electric field at a point R due to dipole at origin 

E -(*fc)£ B| -** |lU 

R is a unit vector along R: R= R/fl. 

Energy of interaction of two dipoles {general case) 

R is the vector from p., to jU^ . 
Energy of interaction of two parallel dipoles 

3cos 2 0) 



(^j/?** 11 



R makes an angle with Pi (andpj). 
Mean dipole moment of freely rotating polar molecule in a 
field E (Langevin function) 

<ti) = p£ (iJE/kTi 
X U) = coth x — -£> tne Langevin function 

W-pfiEmT) kT>liE. 
Molar orien ta tion p olariza tion (Langevin-Debye equation) 

p is the density, a the polarizabiltty, e the relative 
permittivity, M is the molecular weight. 



p-dichtorobenzenes are 1.70 D, 2.25 D, 1.48 D, and zero 
respectively (Fig. D1}. 

The computation of electric dipote moments is a surpris- 
ingly complicated business, and its success depends critically 
on the accuracy of the molecular wavefunctions. The difficulty 
can be appreciated by considering the simplest possible 
approach to their calculation. In this the electron charge 
density on each atom in the molecule, plus the positive nuclear 
charge of the atom, may be represented as a point charge at 
each nucleus. It is then simply a matter of computing the 



dipole moment 



51 




FJG. D1. Vectorial addition of dipole moments illustrated for 1, 2- 
dichlorobenzene. The experimental moment is 2'25 D. 



+qe/2 ^^ ^^+qe/2 

cr X) 





FIG, D2. Contributions to the dipole moment of H^O: (a) point 
charge, (b) asymmetry dipole, (el atomic (hybridization} dipole. 



dipole moment of an array of point charges, distributed 
according to the molecular geometry. This point charge model 
(Fig. D2a) gives bad results because it is a poor approximation 
to assume that all the electronic charge is located at an array of 
points. One factor omitted is the asymmetry dipole. This arises 
from the charge distortion due to the "overlap of orbitals of 
unequal size. Consider the case when an occupied orbital is 
formed from the overlap of a large p-orbital on one atom and 
a small s-orbital on another. The region of maximum overlap 
lies closer to the small atom, and so instead of it being reason- 
able to regard the charge as residing on one atomic nucleus or 
the other, there is a considerable accumulation of charge in 
the overlap region close to the smaller atom (Fig. D2b): the 
asymmetry dipole moment is the contribution to the overall 
moment of this extra charge distribution. It is by no means 
negligible, for it may amount to about 1 D. Another major 
contribution is the atomic dipole, which is also called the 
hybridization dipole and which arises, as the latter name 
suggests, when an electron occupies a 'hybridized atomic 
orbital. When this is so the centroid of electronic charge on an 
atom no longer necessarily coincides with the centre of nuclear 
charge {Fig. D2c). This separation of charge centroids is 
especially important when the hybridized orbital is a "lone 
pair; and the atomic asymmetry it is responsible for is 
reflected in a contribution to the total dipole moment of the 
order of 1 D. The contributions to the total moment of the 
water molecule are illustrated in Fig. D2. 

Methods of measuring molecular dipole moments include 
the Stark effect in molecular spectroscopy and the measure- 
ment of the dielectric constant (electric relative permittivity) 
of solutions (see °polarizability}. 

Questions. 1, What does a non-vanishing dipole moment rep- 
resent tn a molecule? Why is the debye a convenient unit to 
use? Why does a molecule with a centre of symmetry possess 
no dipole moment? From Table 6 estimate the dipole moments 
for o-, m-, andp-dibromobenzene. What are the deficiencies 
of the point -charge model? What is the asymmetry dipole? 
What is its direction? What is the atomic dipole? 
2. The energy of a dipole in an electric field E is determined 
by — )U*E. Caiculate the energy (in J and in J mol -1 ) required 
to reverse the orientation of a dipole of magnitude 



52 



Dirac equation 



1 D{3-3 X 1CT 30 C m) in a field of 100 V m"*. Which orien- 
tation has the lowest energy? Consider a dipole moment on a 
freely rotating molecule; set up the Boltzmann distribution for 
the orientation of the molecule in an electric field, and show 
that the mean dipole moment does not vanish, (In fact, show 
that the mean dipole moment is given by the Langevin 
function {Box 5).) Show that at low applied fields (how low?) 
the mean dipole moment is inversely proportional to £7". Cal- 
culate the degree of ordering, and the energy of the optimum 
configuration, when a polar molecule is in the vicinity of an 
ion. Choose several separations for the species, which may be 
taken to be water and the sodium ion. 

Further information. See Coulson (1961), and Chapter 10 of 
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972) for 
a thorough discussion and examples of calculations, and also 
§8.10. See also Streitweiser (1961) for accounts of the 
computation of dipole moments for organic molecules. Tables 
of moments, their determination and use 3re given by Smyth 
(1955), Smith (1955), Sutton (1955), McCleilan (1963), 
and Minkin, Osipov, and Zhdanov (1970). See §15.18 of 
Moore (1972), and Gasser and Richards' Entropy and energy 
levels (OCS 19) for a simple deduction of the Langevin 
expression. Wheatley (1968) gives a simple introduction to the 
determination and application of dipole moments. The energy 
of interaction between two dipoles, the mean orientation 
polarization (the dipole moment of a collection of freely 
rotating molecules in the presence of an electric field, as 
determined by the Langevin function), and related information 
are given in Box 5. See °polarizability for related concepts. 

Dirac equation. The Dirac equation describes the behaviour 
of an electron in a way that combines the requirements of 
quantum mechanics with the requirements of relativity. The 
trouble with the Schrodinger equation is that it is unsym- 
metrica) in space and time (it contains first derivatives with 
respect to time and second derivatives with respect to space). 
One way out would be to find an equation which had second- 
order time derivatives; but the unfortunate quality of the 
solutions of such an equation (which was proposed by 
Schrodinger and named after Klein and Gordon) is that the 
total probability of the particle being anywhere in the 



universe is predicted to be a function of time; and so the 
equation allows the number of particles in the universe to vary. 
At the time this seemed unacceptable, and Dirac repaired it by 
keeping the equation first-order in time, and, like Procrustes, 
forcing the space derivatives to be first derivatives too. He 
could not do this arbitrarily, and the conditions he was forced 
to impose led him to conclude that he had to deal with wave- 
functions with four components. Two components corre- 
sponded to an embarrassing negative energy, but, instead of 
forgetting the whole thing, Dirac had sufficient confidence to 
propose that all the negative energy states were filled up 
throughout the universe and that we experience only the extra 
particles added to the overlying positive energy solutions. How 
satisfying it was then when it was shown experimentatly to be 
possible to excite the particles out of their negative energy 
states, leaving a hole. Such a hole will have a positive charge 
and positive mass, and is referred to as the positron (e + ). It 
was then shown by Dirac that the two positive -energy sol- 
utions had the same energy, but that when a magnetic field 
was applied this degeneracy was removed, one level rose and 
the other dropped. This can be interpreted in terms of the 
electron having a magnetic moment that can take two orien- 
tations; in other words it is a particle with 'spin ^. The 
magnitude of the splitting was characteristic of a charged 
spin-^- particle, but with a moment twice as large as would 
be anticipated on the basis of a classical model of a spinning 
charge. Thus the anomalous °s-vatue of the electron emerges as 
a natural consequence of the Dirac equation. 

Purifier information. Dirac (1958), of course. A simple 
account of the construction of the Dirac equation is given by 
Moss (1973) and by Bjorken and Drell (1964); see also 
Landau and Lifshitz (1958a), Schiff (1968), Messiah (1961), 
and Schweber (1961) for accounts of increasing sophistication. 
See the Dirac Festschrift, edited by Salem and Wigner (1972), 
for recent developments. 

dispersion forces. The induced-dipole— induced-dipole con- 
tribution to the van der Waals -intermolecular force is the 
London dispersion force. It is a kind of self -generating boot- 
strap force, and arises by virtue of the correlation of fluctu- 
ations in the electron-density distribution of neighbouring 



dispersion forces 



53 



molecules. The fluctuation in the electron density on molecule 
A causes it to possess instantaneously a dipole moment (if it is 
a polar molecule the fluctuation makes an extra ephemeral 
contribution to the permanent dipole). This instantaneous 
dipole polarizes the electrons on a neighbouring molecule; the 
accumulation of electrons on molecule A drives the electrons 
from the neighbouring region of B and so induces there, as in 
Fig, D3, a region of positive charge. Likewise, the region of 
relative positive charge on A matches the region of negative 
charge it induces in B. Such a correlation of charge dis- 
tributions lowers the energy and corresponds to a binding 
force. Note that it is essential that there be a correlation 
between the charge fluctuations, for otherwise the effect 
would disappear. 



driving for response) 
.fluctuation 




response (or driving) 
fluctuation 

FIG. D3. The induced-dipole— induced-dipole (dispersion) interaction. 



The dispersion energy has a characteristic ft" 6 dependence 
{which is reflected in the Lennard-Jones 6-, 12-°intermolecular 
potential} whose source is easy to detect in the model: the 
field arising from the initial dipole on A has an ft -3 depen- 
dence {see Box 6), and the return interaction, where the dipole 
on B interacts with the dipole already on A, contributes 
another ft" 3 . The interaction energy also depends upon the 
"polarizability of the molecules, for it is a measure both of 
the ease with which an electric field may distort the molecule 
and of the extent of its charge fluctuations. The energy of 
interaction of two spherical molecules separated by a distance 
ft and with polarizabilities a and a and ionization potentials 
/ and / B is given by the London dispersion formula, quoted 
in Box 6. Its magnitude is of the order of a few kJ mol~ l . The 
-ionization potential enters the formula because the extent of 



BOX 6: Dispersion energy between spherical molecules 
Energy of interaction of a and b separated by ft: 

X [d{a) 0n .d(a) mo ][d(b) 0n .d(bU]. 
where A mn = (E m - E g ) a + (jF ffl - E) b the excitation 
energies of a and b, 

d 0m (a) = S&npS < a > d ^ m (a), the transition dipole of a, 

don <b> = /d7^S(b)d^ (b), the transition dipole of b. 
London formula 

where a is the polarizability of the atom and / is its 

"ionization potential. 

If a is in cubic aVigstroms, ft in aVigstroms, and / in elec- 

tronvolts. 

"•-WWW- 

3/327T 2 - 0-0095. 
Retardation. When ftA/hc > 1 



fluctuation depends on the ease with which a molecule can be 
excited. 

At great distances the dispersion formula fails on account 
of the time it takes the information that a fluctuation has 
occurred to travel between the atoms: this is the retardation 
effect. When ft is large (ft >- fie/A, A a typical molecular 
excitation energy) the ft -6 dependence is replaced by ft" 7 , and 
the interaction falls off more rapidly that the London formula 
predicts. 

Questions. What are the dispersion forces, and how do they 
arise? What is the dependence on the separation of the mole- 
cules? Why does the interaction energy depend on the 
polarizability of the molecules? On what other properties does 



54 



doubling 



it depend? What does the distance-dependence become at large 
distances, and what is the change due to? Why are hydro- 
carbons volatile? Why does l 2 dissolve readily in liquid 
ammonia? Why is it difficult to liquefy helium? 

Further information. See MQM Chapter 1 1 for the deduction 
of the quantum-mechanical expression for the dispersion 
interactions, and of the London formula. See also Kauzmann 
(1957), Chapter 18 of Ey ring, Walter, and Kimball (1944), 
Chu (1967), | Ml of the book of Hirschf elder, Curtiss, and 
Bird (1954). A brief survey has been given by Longuet-Higgins 
(1965); see also Rowlinson (1969) and Curtiss (1967). The 
subject of -intermoiecular forces is reviewed under that 
heading, and more references will be found there. 

doubling. In a spectrum 'doubling' refers to a situation in 
which the -degeneracy of a level is removed by some hitherto 
neglected interaction; each two-fold degenerate level is split 
into two distinct nondegenerate levels— hence 'doubling'. The 
doubling shows up in the spectrum either as an actual doubling 
of the number of lines, when other states can make transitions 



L 



U ' 




V 


V \ 






b 


i 



























FIG. D4. Different types of doubling in a spectrum: (a) the originally 
degenerate level combines with one level; (b) the originally degenerate 
level combines with two different levels. Unperturbed levels are in 
black, perturbed levels are in colour. 



FIG. D5. A-doubling: the mechanism in a diatomic molecule. 

to the now separated levels (Fig. D4a), or there occurs a situ- 
ation in which some states combine with (which means make 
transitions to) only one of the components of the doublet, and 
others combine with the other component (Fig, D4b). 

An example of doubling in atomic spectroscopy is the •fine 
structure in the spectra of the aikali metals. Take sodium as an 
example, and consider an excited configuration in which the 
valence electron occupies a 3p-orbital, then drops down into 
the 3s-orbital, and in the process emits the characteristic 
yellow sodium light (589 nm). Close inspection of the 
spectrum shows that the yellow light is slightly impure, for it 
consists of two components, one at 588.99 nm and the other 
at 589.59 nm; this splitting can be ascribed to a doubling of 
the excited-state levels. More details will be found under -fine 
structure; but, briefly, the excited state is 2 P, this has two 
'levels 2 Pi/2 and 2 P3/2, which are degenerate in the absence of 
spin-orbit coupling but differ in energy when the coupling is 
taken into account. Transitions from these two levels to the 
ground state 2 S I/2 yield the two spectral lines separated by 
17-2 cm" 1 . 

An example of doubling in molecular spectroscopy is 
/^doubling (lambda doubling) in which the rotation of a 
linear molecule removes the degeneracy of the two components 
of a II "term (or, less importantly, of a A or a <& term). It is 
possible to identify the interaction responsible by considering 
a diatomic molecule in which a single valence electron occupies 
a Jr-orbital. In Fig, D5 we illustrate the situation in which the 
electron occupies either the orbital tt or the orbital if , where 



doubling 



55 



the superscript is based on the behaviour of the sign of the 
orbitals when they are reflected in the plane of molecular 
rotation. If the electrons were able to follow the rotation of 
the nuclei exactly, with no lag or slip, the two TT-orbitals 
would have precisely the same energy; this is well-known to be 
the case when a static molecule is considered, as in the usual 
discussion of bonding theory. But the electrons do not follow 
the motion of the nuclei exactly (see "electron slip and the 
"Bom-Oppenheimer approximation), and there is a tendency 
for the electrons to lag behind the nuclei. In the case of if the 
lag has virtually no effect, for to the lowest approximation the 
nuclei slide a little forward in the orbital's nodal plane and so 
remain surrounded by roughly the same electron density as in 
the static case. But the nuclei slide from the vertical nodal 
plane of 7T + into a region of non-zero electron density, and so 
it is entirely plausible that the energy of ?r + is modified by the 
rotation. It follows that the two orbitals move apart in energy 
and that the energy separation, or doubling, depends on the 
rate of rotation. 

The effect of A-doubling on a spectrum may be seen by 
considering the "rotational structure of an "electronic trans- 
ition in which a diatomic molecule changes from a S state to a 
Ft state. (For the technical, we are considering the transition 
l II«— '2 + .) The A-doubling of the upper state is illustrated 
in Fig. D6: the separation of the components of the state 
increases with "rotational quantum number J according to the 
ru\eqJ{J + 1 ), where q is a constant. Transitions of the P- and 
R- "branches of the spectrum are allowed to take place from 
the rotational states of the lower electronic state to only one 
of the components (the one labelled +) of the upper electronic 
state, but the Q-branch transitions may take place only to the 
other component. From an analysis of the shift of the Q- 
branch from the centre of the P- and R -branches it is possible 
to extract the value of the parameter q (see Questions) and to 
relate this value to the separation of electronic states in the 
molecule. 

Two other examples of doubling are important. The first is 
il-type doubling, in which the degeneracy of a bending vi- 
brational level of a linear triatomic molecule is removed by 
the effects of molecular rotation (see "Coriolis interaction). 
The second is 'inversion doubling, where the abi I ity of a mole- 
cule to invert from one conformation to another doubles its 



vibrational spectrum. Qrdoub/ing (omega doubling} is the 
analogue of A-doubling when -Hund's case (c) is the appro- 
priate coupling scheme. 

Questions. 1. What is the meaning of the term 'doubling'? 
How does its presence show in the spectrum of an atom or 



J 

4- 














* < 




« 4- 




T 








E2q 


2 + 








1 












TT i + 












4 












I 


3 












2 










1 






z* ° 





p 












rR 






1 
1 


1 ! 
! i 

! 






L> 




Q- 1 


1 I 









FIG. D6. A-doubling: selection rules and spectra. A large A-doubling 
of the upper state has been selected. Compare Fig. B12, which is based 
on the same parameters. 



56 



duality 



molecule? Discuss the doubling that .arises in the spectra of the 
alkali metals: what is the interaction responsible for the 
doubling; how does the doubling change as the atom is 
changed from Na to Cs; are the "selection rules satisfactory for 
the explanation? If the two lines are at 588.99 nm and 589.59 
nm, what is their separation in cm" 1 ? Deduce a value of the 
spin-orbit coupling parameter of Na from these data. What is 
the origin of A-doubling? Why should the doubling increase 
with rotational quantum number? By inspection of Fig. D5 
deduce that the orbital mixed into the 7T + -orbital by the 
effect of rotation is a fJ-orbital. Is the orbital mixed in a 
bonding or an antibonding orbital? What mixing is expected in 
the case of an "antibonding 7r-orbital? (It should be stressed 
that the actual calculation of A-doubling effects should be 
done on the basis of overall states rather than one-electron 
orbltals.) 

2. The A-doubling interaction varies as qJ{J+ 1 ); show that in 
the spectrum of a diatomic molecule the R- and P -branches 
correspond to a diatomic molecuie with a modified rotation 
constant B +b, where B is the true rotational constant, and the 
Q-branch corresponds to a molecule with a different rotational 
constant B+b . Find expressions for b and b' in terms of q 
and show how the displacement of the branches may be used 
for the determination of the true B. Construct the first few 
lines of the rotational structure of the electronic transition in 
which a '£* molecule changes to 'II; in the lower state take 
B = 6020 cm" 1 , in the upper B = 6-000 cm" 1 , and in the 
upper q = 0008 cm -1 . 

Further information. See MQM Chapter 10 for a more detailed 
discussion, Whiff en (1972), and Barrow {1962). A detailed 
discussion of A-doubling with worked examples is given in 
Herzberg (1950): see especially pp. 252—5; the basic theory is 
laid down on p. 226. Sugden and Kenney (1965) discuss 
ikype doubling in § 3.1c, and King {1964) gives a helpful 
discussion of A-douhling in § 6.7, fi-type doubling in § 9.15, 
and inversion doubling in § 9.17. The detailed algebra of 



A-type doubling is described by Kovacs (1969). See also 
Townes and Schawl ow (1955), 

duality. Classical physics dealt with waves and with particles, 
and it was quite clear which was which. Quantum mechanics 
introduced the view that matter and radiation have a dual 
character and that either aspect of behaviour may be exhibited 
by the same entity. Thus what for historical reasons had been 
classified as particles showed behaviour that hitherto had been 
characteristic of waves: it was discovered, by Davisson and 
Germer, that crystal lattices could diffract beams of electrons, 
and Thomson showed that the same effect could be brought 
about by passing the beam through a thin gold foil. Conversely, 
it was discovered that what had been classified for historical 
reasons as waves showed behaviour that hitherto had been 
characteristic of particles: both the "Compton effect and the 
•photoelectric effect require the energy or momentum of the 
radiation to move around in localized bundles. The conclusion 
to which one is forced is that 'waves' and 'particles' have been 
so classified because at the level of the experiments done until 
the beginning of the present century one type of behaviour 
dominated and the other was concealed. Looking more closely 
at each type of entity reveals the duality of their character, 
and so the nature of matter and radiation is neither just one 
nor the other, but a composition of the two. Which aspect 
dominates depends on the experiment; no experiment can 
exhibit both aspects of the duality simultaneously (principle 
of -complementarity). 

Further information. See Jammer { 1 966) for the way that the 
idea of duality emerged as a fundamental property of matter, 
and Bohm (1951) for an illuminating discussion. The quantum- 
mechanical basis of duality may be investigated by dealing 
with -wave packets and studying the role of the "uncertainty 
principle in limiting our view of Nature. We have mentioned 
the duality of matter and radiation; further aspects will be 
found under °excitons, -phonons, "photons, and "polarons. 



E 



eigenf unctions, etc. When some mathematical operation 
(such as multiplication, division, or differentiation) is done on 
a function the result is generally soms different function. Thus 
differentiation of the function x 2 yields the different function 
2x. But some combinations of operations and functions are 
such that when the operation is done the same function is 
regenerated, but perhaps multiplied by a number. Thus 
differentiation of the function exp 2x gives 2exp 2x, which is 
the same function multiplied by the number 2. When this 
occurs the function is referred to an an eigenfunction of the 
operator (in this case the differential operator d/dx), and the 
numerical factor (2 in the example) is called the eigenvalue of 
the operator. Eigen is the German word meaning 'own' or 
'particular'. 

The Importance of these names in quantum mechanics can 
be recognized by noting that the Schrddinger equation may be 
written as Hip = E^/, where H is a differential operator {the 
"hamiltonian) and \p is the °wavefunction. This has the form 
of an eigenvalue equation, with the energy E playing the part 
of the eigenvalue and the wavefunetion as the eigenfunction. 
The wavefunction represents a state of the system, and so \p 
is often termed the eigenstate, or, because it also shares some 
properties with vectors, the eigenvector. The result of an 
experiment done on a system in a given eigenstate is the 
eigenvalue of the "operator corresponding to the observation. 

Questions. 1. Define the terms eigenfunction, eigenvalue, 
eigenstate, eigenvector, and eigenvalue equation. Why are these 
terms important in quantum mechanics? If you knew the state 
of a system, and knew the mathematical form of the operation 



corresponding to the observable you wanted to determine, 
how could you predict the result of an experiment? 
2. Which of the following functions are eigenfunctions of the 
operation d/dx : ax, ax 2 , exp ax, exp ax 2 , In ax, sin ax? Which 
are eigenfunctions of d 2 /dx 3 ? The operator corresponding to 
the component of angular momentum about thez-axis is 
(h/i) {3/3 0): if you knew that the state of the system is 
described by the function f[r, 0)exp im0, what experimental 
result could you predict? What if the state were f[r, 6*)cos m0? 
Here is a tricky but important question: what is the eigen- 
function of the operation 'multiplication by x'f 

Further information. See MQM Chapter 4 for a discussion of 
operators and eigenvalues. For a deep view of eigenstates and 
an explanation of why they are called eigenvectors see Dirac 
(1958), von Neumann (1955), and Jauch (1968). See Davydov 
(1965) for a helpful summary and examples, including {on 
p. 28) the answer to the final question, 

Einstein A and B coefficients, "ihe rate of an absorptive 
transition between two states is proportional to the density of 
radiation present at the frequency of the transition, and the 
coefficient of proportionality is the Einstein coefficient of 
stimulated absorption B. The rate of emission depends on two 
factors. One contribution is proportional to the energy density 
at the transition frequency, and the coefficient of proportion- 
ality is termed Einstein's coefficient of stimulated emission. 
Since it is equal to B it is also denoted B. Another component 
of the transition rate is independent of the amount of radi- 
ation already present and is termed Einstein's coefficient of 
spontaneous emission and denoted A. 



57 



58 



electric dipole transition 



The necessity for this extra coefficient A stems from the 
equality of the rates of upward and downward stimulated 
transitions: if these stimulated processes were the only ones to 
occur the distribition of populations between the upper and 
lower states would be equal at equilibrium, whereas the 
Boltzmann distribution requires the upper population to be 
less than the lower. The spontaneous emission process comes 
to the rescue because it provides a way of permitting the 
upper states to leak into the lower at a rate independent of 
the radiation already present. Thus if the equilibrium popu- 
lation of the upper state is fJ + , and of the lower is n_, then 
the rate of upward, absorptive, transitions is/7 _Bp(U), where 
p(v) is the radiation density at the frequency of the transition 
of energy/)!', and the rate of the downward transitions is 
n t Bp{v) + n t A. Equating the two rates at equilibrium, and 
insisting with Boltzmann that n in_ = exp {—hv/kT), enables us 
to deduce that the equilibrium radiation density must be of the 
formp(p) = {A/B} [exp {hv/kT)— 1]~' . Simply looking at this 
expression shows that ,4 cannot be zero if the equilibrium 
radiation density is to be non-zero, and so the spontaneous 
process is essential. But one can go even further, and find an 
expression for A in terms of B by recognizing that at equilib- 
rium the radiation density will be that of a °blaek-body 
radiator, and therefore that p should be given by the Planck 
distribution. By comparing the expression above with that in 
Box 2 on p. 23 it is easy to deduce that A = 87tf){f/c) 3 S. 

An expression for B may be obtained from "time-dependent 
perturbation theory, and for -electric dipole transitions one 
finds that B = d 2 /6e h 2 , where d is the magnitude of the 
-electric transition dipole between the two states of interest. 

An important conclusion emerges from the expression for 
the ratio A/B, for we see that it increases as the cube of the 
frequency of the transition; therefore the spontaneous 
processes become very important at high frequencies, but can 
be ignored at low frequencies. A consequence of this is the 
inherent difficulty of constructing high-frequency "lasers, for 
these rely on the co-operative effect of stimulated transitions; 
if the spontaneous processes are too important the excited 
states decay too quickly and do not contribute to the co- 
operative laser process. 

Questions. 1. What is the significance of the Einstein B 
coefficient? What is the rate of transition from a ground 



state occupied by N atoms? What is the rate of transition 
from an upper state occupied by N" atoms? If A were absent, 
show that the equilibrium populations of the upper and lower 
levels are equal and in conflict with the Boltzmann requirement. 
Deduce an expression for the Einstein A coefficient on the basis 
of a Boltzmann distribution of populations, equal rates of 
transition at equilibrium, and the Planck distribution of 
radiation at equilibrium (p. 23). At what frequencies are the 
spontaneous emission processes important? Confirm that the 
dimensions of B are those of (volume X frequency) per 
(energy X time), and that since the dimensions of p are energy 
per (frequency range X volume) the transition rate comes out 
with the correct dimensions. The dimensions of A should be 
those of time" 1 ; check that this is so. 
2. From the Planck distribution (Box 2, p. 23} estimate the 
radiation density of light of wavelength 1 cm (microwave 
radiation) and 500 nm (visible light) emitted by a biack body 
at 1000 K; deduce a value of 8 that corresponds to unit 
'oscillator strength and compute the relative importance of 
the emissive and absorptive processes at equilibrium. 

Further information. See MQM Chapter 7. A discussion of the 
A and B coefficients is given in § 12.8 of Davidson (1962), 
in § 8d of Eyring, Walter and Kimball (1944), in § 7.2 
of Hameka (1965), in Chapter 5 of Heitler (1954), and, 
implicitly, by Dirac (1958). Their importance in the discussion 
of laser radiation processes is reflected in books on laser 
technology; therefore see "laser and references therein. 

electric dipole transition. The most intense transitions of 
atoms and molecules are those caused by the interaction of 
the electric component of an electromagnetic field with the 
electric dipole moment of the system. A simple example is the 
interaction of a permanent dipole moment of some molecule 
with the light beam; when this happens the rotational motion 
of the molecule is accelerated and it makes a transition 
between two "rotational energy levels. Conversely, a rotating 
electric dipole moment behaves like an oscillating electric 
charge, and an electromagnetic wave is generated and trans- 
mitted, by the rotating system. The wave carries away energy, 
and so the molecule drops down the ladder of rotational 
energy states. The same kind of electric dipole transitions 
occur in vibrating systems if accompanying the vibration there 



electric dipole transition 



59 



is a net motion of charge. Thus electric dipoJe transitions can 
be induced in the "vibrational energy levels of the polar HCI 
molecule because an observer on the side of the molecule will 
see a changing charge distribution as it vibrates. The same 
observer viewing the chlorine molecule (Clj ( will detect no 
such change, and so that. mode of the molecule cannot interact 
with the electromagnetic field by an electric dipole mechanism; 
it can neither accept energy by absorption nor generate energy 
in the field and emit. 

Electric dipole transitions may also occur between the 
electronic states of atoms, and once again it is necessary to 
assess whether the transition involves a motion of charge 
which can be interpreted as an oscillation of a dipole. It should 
be clear that a transition from a spherically symmetrical 1s- 
orbital to a spherically symmetrical 2s-orbital involves a sym- 
metrical redistribition of charge, and so no net dipole moment 
can be associated with the transition. Therefore this transition 
cannot occur via an electric dipole-moment mechanism. If 
however we envisage an s-to-p transition, then a dipole 
can be identified. If the transition is from s to p the oscil- 
lating dipole lies along thez-axis, and so the emitted radiation 
is transmitted plane-pofarized with its electric vector in the 
z-direction (the intensity would appear greatest to a viewer 
stationed in the xy-plane). If the transition is from s to p 
(or p ) the polarization lies in the direction of x (or y), and 
the radiation emerges in a belt largely in the yz (orzx)-plane. 

When transitions are made between orbitals with definite 
values of the 'quantum number m^ the radiation is circularly 
polarized; when m^ decreases by unity the emitted radiation is 
left circularly polarized, and when it increases by unity the 
radiation is right circularly polarized; when mj is unchanged 
the light is plane polarized. (We use the convention that the 
electric vector of left circularly polarized light rotates anti- 
clockwise to the observer towards whom the light is travelling.) 
This behaviour can be understood by considering the trans- 
itions s— p; p^ and p may be expressed as superpositions of 
states of different mg, and so the radiation emitted is the 
appropriate superposition of the plane polarized radiation 
already described. 

The "selection rules for electric dipole radiation may be 
understood in terms of the above discussion and also by virtue 
of the "photon's possession of an intrinsic "spin angular 



momentum. The first view leads to the Laporte selection rule 
which states that the parity of an orbital (see "gerade and 
ungerade) must change in an electric dipole transition. Thus in 
an s— p transition an even orbital (even with respect to 
inversion through the atomic nucleus) changes to an orbital 
odd under inversion. The same is true of d— p transitions. The 
rule forbids s— s, p— p, and d— d transitions, for with these 
there is associated no transition dipole moment. The existence 
of a photon angular momentum coupled with the principle of 
conservation of angular momentum provides an alternative 
view of the situation, because the labels s, p, and d imply that 
electrons in the orbitals possess different "angular momenta, 
and when a photon is emitted or absorbed the angular 
momentum of the atom must change in order to conserve the 
total momentum. A photon impinging on an s-orbital must 
turn it into a p-orbital, and is unable to turn it into a d-orbital 
because it brings insufficient angular momentum. When an 
incident photon interacts with a p-orbital the final state can be 
either a d-orbital or an s-orbital, depending on the relative 
orientation of the angular momenta in the collision. 

The connexion between the change in the value of mn and 
the polarization of the light can also be understood on the 
basis of the photon spin, because left circularly polarized light 
corresponds to one orientation of the photon spin with respect 
to its propagation direction and right circularly polarized light 
corresponds to the opposite projection. Therefore in order to 
conserve the total amount of angular momentum of the total 
system about a particular direction, nta must change appro- 
priately if a photon is absorbed or emitted. If the light 
approaches the atom along the negative z-ax is (from z = — °°) 
and is left circularly polarized (spin projection +1 on the 
propagation direction, moving like a right-handed screw) 
when it is absorbed the atom must change from a state mg to 
rr)g+1 if the angular momentum about thez-axis is to be 
conserved. The same argument applies to emission. 

If an optical transition is observed when the preceeding 
rules forbid it, the reason may be that the atom lacks a centre 
of symmetry by virtue of its environment, or there may be a 
coupling between the electronic and vibrational modes: see 
-vibronic transitions. Or it may be that a "magnetic dipole or 
electric quadrupole transition is responsible for the intensity. 
To a very good approximation the electric field of the light 



60 



electron affinity 



cannot interact directly with the spin of an electron, because 
that is an internal mode of motion, and so spin is conserved in 
electric dipole transitions; when that rule fails it is by virtue 
of the spin-orbit coupling interaction. 

Questions. 1. What are the most common and intense optical 
transitions due to? Why is an electromagnetic field able to 
accelerate a rotating molecule? To do so what must the mole- 
cule possess? Can an electric dipole transition increase the 
amount of angular momentum about the figure axis (the sym- 
metry axis) of a symmetric top molecule (a cylindrical or disc- 
like molecule)? Is the light polarized in rotational electric 
dipole transitions? Under what circumstances can a vibrational 
mode emit or absorb light by electric dipole transitions? Which 
modes of vibration of carbon dioxide are able to interact with 
an electromagnetic field, and which are not? (It might be help- 
ful to revise "normal modes and "vibrational spectra.) Why is it 
possible to regard the transitions between states of atoms as 
involving an electric dipole moment? What is the name given 
to the moment involved? How is the polarization of the light 
involved in a transition related to the orientation^ the 
electric dipole? Discuss the polarization of the transitions 
s-p ; , s-p x , s~{p x + \p y ), where by (p^ +ip ) is meant the 
p-orbital with m% = + 1. Consider the effect of a magnetic 
field on an atom, as in the °Zeeman effect: discuss the polar- 
ization of the light that you would see as your detector, 
sensitive to the three transitions s— p s— p +[ , s— p t , was 
moved over the whole sphere of possible orientations. State 
the Laporte selection rule: how can it be justified and under 
what circumstances will it fail? What role does the spin of a 
photon play in determining which electric dipole transitions 
are allowed? How may such a description be brought into line 
with the Laporte selection rule? Discuss the polarization of 
the transitions in the preceeding questions in terms of the 
angular momentum of the photon and the conservation of 
total angular momenta. Why may the angular momentum of 
an atom either increase or decrease when a photon is absorbed 
or emitted? If electric dipole transitions forbid a transition, 
what might account For its intensity? 
2. Discuss the sense in which an electric transition dipole 
moment may be said to oscillate at the frequency of the 
radiation it transmits (the 'correspondence principle requires 



some such sense to exist). Consider the time-dependence of 
the transition dipole moment by expressing the matrix 
element fdr^d^ , where ty is the excited state, ^ is the 
ground state of the system, and d is the electric dipole moment 
operator, in terms of the explicit time-dependence of the 
states; that is, write ^ = \jj exp(— if f/h). Apply group 
theory {see "character) to the matrix element to deduce the 
Laporte selection rule. 

Further information. See MQM Chapter 8 for a discussion of 
electric dipole transitions. For applications in organic chemis- 
try see Murrell (1971) and Sandorfy {1964), and for appli- 
cations in inorganic chemistry, especially in the electronic 
spectra of transition-metal compounds see Orgel (1960), 
Ballhausen (1962), and Griffith (1964). The basis of the 
selection rules is the calculation of transition probabilities by 
time-dependent perturbation theory: therefore see MQM 
Chapter 7, Herzberg (1940), and Eyring, Walter, and Kimball 
(1940). 

electron affinity. The electron affinity of an isolated 
atom is the energy evolved when an electron is brought up 
from infinity and the anion formed. Therefore the electron 
affinity is the difference in energy of the neutral atom and 
its negative ion: E (X) = £(X)— £(X~). A positive electron 
affinity implies that the anion is more stable than the neutral 
atom. Since an amount of work E (X) must be done on the 

a 

ion X" in order to regain the neutral atom and the infinitely 
separated electron it should be clear that E (X) is the same as 
the "ionization potential /(X~) of the anion: E (X) = /(X"). 

The determination of electron affinities is often some- 
what devious. A direct method is to measure the ionization 
potential of the anion. The other methods are: 

{1 ) electron impact, in which one tries to identify the 
appearance potential of the negative ion in the reaction 
XY + e -> X" + Y; 

(2) electron attachment measurements (so-called electron 
affinity spectroscopy); 

(3) polarography; 

(4) application of the Bom-Haber cycle for the lattice 
energy of an ionic crystal. 

One application of electron-affinity values is to the construe- 



electron slip 



61 



tion of the Mulliken 'electronegativity scale, and it is of use 
wherever the stability of an anionic species is required. 

Further information. A review of methods for determining 
electron affinities has been written by McDowell (1969). 
Further information and tables of affinities will be found in 
Prichard (1953), Vedeneyev, Gurvich, Kondrat'yev, 
Medaredev, and Frankevtch (1966), and Briegleb (1964). For 
applications in chemistry, see Puddephatt's The periodic table 
of the elements {OCS 3) and Phillips and Williams (1965). 

electron slip. Electrons slip not because they are slippery but 
because in a rotating molecule they may be unable to follow 
the rapid motion of the nuclei: electron slip is a manifestation 
of the breakdown of the "Born-Oppenheimer approximation. 
Think of a rotating hydrogen molecule: the electrons may lag 
behind the motion of the nuclei as the nuclear framework 
rotates. That is, the electrons and the nuclear framework have 
slightly different angular momenta. 

There are two important consequences of electron slip. The 
first is that the distinction between O and U orbital s (and 
between S and Fl states) becomes blurred: this is manifest in 
the electronic spectra of linear molecules as A-type "doubling. 
The other consequence is that all molecules possess a magnetic 
moment by virtue of their rotation: this is the molecular 
magnetic moment. This moment can be traced to the different 
rates of rotation of the positively-charged nuclear framework 
and the negatively-charged electron cloud of the molecule. 
There is a net current, and therefore a net magnetic moment. 
All molecules possess such a moment when they rotate, and the 
magnitude of the moment increases with the "rotational 
quantum number J, 

Further information. See MQM Chapter 10 for a more detailed 
account of the way that electron slip mixes 1> and II states, 
and -doubling for a picture; for a quantitative treatment see 
King (1962), Herzberg (1950), and Kovac's (1969). For mole- 
cular magnetic moments see "y-value, §11.6 of Townes and 
Schawl ow (1955) and p. 299 of Herzberg (1950). 

electron spin resonance: a synopsis. The electron spin 
resonance (e.s.r.) experiment is the observation of the energy 
required to reverse the direction of an electron -spin in the 



presence of a magnetic field. The electron possesses a 
"magnetic moment by virtue of its spin, and in the presence 
of an applied field the two permitted orientations (or and 0) 
have different energies (in a fietd of 3 kG thecc-state lies 
0-3 cm" 1 above thep*-st3te). An electron can be induced to 
reverse its orientation (make a transition from |8 to a) if 
electromagnetic radiation of the appropriate frequency is 
applied, and in a 3 kG magnetic field the 0-3 cm" 1 radiation 
(wavelength 3 cm, frequency 9 GHz) lies in the microwave 
region of the spectrum. The apparatus therefore consists of a 
magnet capable of providing a homogeneous field in the 
vicinity of 3 kG, a source of 3 cm microwaves (a klystron), 
and a device for detecting whether the incident radiation is 
absorbed. The e.s.r. experiment is performed by maintaining a 
constant microwave frequency and sweeping the applied field 
until the incident radiation is absorbed (Fig. El ): at this field 
the separation between the a and jS orientations exactly 
matches (is in "resonance with) the radiation frequency. The 




FIG. El, The electron spin resonance transition, and the resulting 
spectrum. 



electron spin resonance: a synopsis 



sample, which must be paramagnetic, may be a solid, a liquid, 
or (more rarely) a gas. 

Three principal pieces of information emerge from the 
experiment: 

1 . The position of the spectrum. The magnetic field experi- 
enced by the electron might differ from the applied field 
because the latter is able to induce local fields. For a given 
microwave frequency the resonance condition will be attained 
at the same local field, and therefore at slightly different 
applied fields in different species. If the microwave frequency 
is V, so that each photon carries the energy hv, and the applied 
field is B, the resonance condition is gfi 8 = hv, where g is a 
factor (which, by lack of inspiration, is called the ^-factor) 
which takes into account the possibility that the local field is 
not exactly equal to B. Measuring the position of the spectrum 



f-eld 




00 60 




FIG. E2. The source of hyperfine splitting in a radical containing one 
spin-% nucleus (denoted by the larger circle}. 



FIG. E3. A typical electron spin resonance spectrum of a radical (with 
one spin-1 nucleus) trapped in a crystal, Two orientations of the crystal 
are shown: note that the centre of the spectrum shifts (9 anisotropic} 
and the splittings change (anisotropic hyperfine interactions). 

enables g to be determined, and as g depends on the electronic 
structure of the paramagnetic species some deductions may be 
made about it. In organic and inorganic radicals g is most use- 
ful for the identification of the species; wider variations in its 
value are found in transition-metal ions, and there it can give 
useful structural information, particularly about the separation 
of energy levels and the spread of electrons on to ligands. g 
may be anisotropic, in which case the position of the spectrum 
depends on the orientation of the paramagnetic species {when 
it is trapped in a crystal, and the crystal rotated, the resonance 
position changes). 

2. The hyperfine structure of the spectrum. Generally the 
spectrum does not consist of a single line: the structure ob- 
served is due to the "hyperfine interaction of the electron and 
any magnetic nuclei present. A magnetic nucleus (such as a 
proton) gives rise to a local magnetic field, which, depending 
on the relative orientation of the nuclear spin and the applied 
field, can increase or decrease the local fieid experienced by 



electron spin resonance: a synopsis 



63 



the electron spin. This implies that radicals with different 
nuclear spin orientations resonate at different applied magnetic 
fields, and the spectrum from a large collection of radicals con- 
sists of lines at all these applied fields (Fig. E2). For example. 
N radicals each containing one proton will constitute a sample 
consisting of jN radicals with proton spins aligned along the 
magnetic field and ~N proton spins aligned against it. The 
electrons in the first set of radicals experience one local field 
and resonate at the appropriate applied frequency, and the 
electrons In the other set resonate at another applied field. The 
spectrum therefore consists of two lines separated by several 
gauss: this 'hyperfine splitting {h.f .s,} may be interpreted in 
terms of the probability that the unpaired electron will be 
found in the vicinity of the magnetic nucleus in question, and 
so a study of the h.f.s. enables the electron distribution to be 
mapped over the molecule. The h.f.s. has both isotropic and 
anisotropic components (see Fig, E3): the former is due to the 
"Fermi contact interaction and is characteristic of s-orbital 
character of the electron, the latter is due to the dipole-dipole 
interaction and is characteristic of p-orbital character. Therefore 



$ 




-1 




f 




JO 



proton spin: 

• a 

o(3 



FIG. E4. A typical solution electron spin resonance spectrum (of 
benzene - ), and its interpretation. 




hyperfine 
interaction 




FIG. E5. The spin-polarization mechanism in CH, 

a study of the angular dependence of the h.f.s. yields infor- 
mation on the "hybridization of the unpaired electron's orbital, 
and this can be used to discuss the shape of the radical {the 
methyl radical, for instance, is shown to be planar). In fluid 
solution only the isotropic h.f.s. is observed (see, for example, 
Fig. E4), and a principal application is its use to map spin 
density in organic radicals. In aromatic radicals the h.f.s. is due 
to a spin-polarization mechanism along a C— H bond, as illus- 
trated in Fig. E5. The jr-electron (with, we choose, O-spin) 
causes the a-spin in the C— H a-bond to be predominantly in 
its vicinity (electrons prefer to be parallel in atoms: see the 
-Hund rules) so that the j3-spin predominates in the vicinity of 
the proton. Therefore, although in the bond the spins are 
paired, the proton sees predominantly one spin, and there is a 
net interaction. 

3. The shape of the spectral tines. The shape of the lines 
depends on the type of motion that the radicals undergo, for 
it is determined by relaxation processes. Two types of relax- 
ation process may be distinguished: the spin-lattice and the 
spin-spin. The former arises by virtue of the motion of the 
radical giving rise to fluctuating magnetic fields at the unpaired 
electron; if these fluctuations happen to have a component 
that oscillates at the transition frequency, then a transition can 
be induced. The lifetime of the upper state is shortened, and, 
by what passes for the "uncertainty principle, the energy of 
the state is blurred. This spin-lattice relaxation process is 
feeble when the motion of the jitterbugging molecule is slow 
(because there are no oscillations in the fluctuating field with 



64 electron spin resonance: a synopsis 

1 

anisotropics dominant 



width 




anisotropics 
' overaqed 
out 



mobility of radicals 

FIG. E6. The dependence of the relaxation time on the rapidity of 
molecular motion. 

the correct frequency 10'° s" 1 ), passes through a maximum 
when a typical time scale for molecular motion is 1Q~'°s, and 
then declines in extremely mobile liquids when many of the 
fluctuations occur at very high frequencies (Fig. E6), The 
lifetime of the spin state is called 7"i , the spin-lattice relaxation 
time, or the longitudinal relaxation time. The longer 7"] the 
narrower the line. 

The other relaxation process is the spin -spin process: in a 
fluid but viscous solution each radical is in a particular mag- 
netic environment or at a particular orientation, and so the 
local fields are all slightly different; consequently the radicals 
resonate at slightly different applied fields, and the spectrum 
is a collection of broadened lines. As the motion of the mole- 
cules increases the differences in the magnetic environments, 
or the anisotropic interactions, are averaged out, and the lines 
narrow. The effect of this broadening process is characterized 
by the spin -spin relaxation time, or the transverse relaxation 
time, T 2 : the longer T 2 , the narrower the line. The broadening 
effect disappears as the radical mobility increases (by raising 
the temperature or reducing the viscosity). 



Note that only T x reduces the number of radicals in the 
upper spin state, and so only it is a true energy relaxation 
mechanism (the direction of energy flow, which is out of the 
spin system and into the lattice, or environment, is determined 
by the thermodynamics of the system: the entropy increases 
for heat flow out of the small spin system into the virtually 
infinite lattice). The T 2 process is a different kind of relax- 
ation process: it relaxes the relative phase of the precessing 
electron spins (Fig. E7). 

A study of the line widths and saturation behaviour (the 
reduction of intensity at high microwave powers as a result of 
the equal ization of the a and jS populations) of the spectrum 
gives information about the motion of molecules in liquids, 
because details of the rapidity of the motion can be inferred 
from the shape of the lines. Information can also be obtained 
about the rate of chemical processes, for example tautomer- 
ism, because these also modulate the environment of the 
electron spins. 

Further information. See Magnetic resonance by McLauchlan 
(OCS 1 ) for a simple account of the principles and applications. 




FIG. E7. The relaxation processes characterized bv T\ and T%. 



electronegativity 



65 



See also Lynden-Bell and Harris (1969), Carrington and 
McLauchlan (1967), Wertz and Bolton (1972), Slichter (1963), 
Abragam and Bleaney (1970) for general accounts. Instrumen- 
tation is described by Poole (1967). The application of e.s.r. to 
organic systems is described in some of these and in Ayscough 
(1967); the application to inorganic, non-transition -metal 
systems is described in Atkins and Symons (1967), the appli- 
cation to transttton-metal ions is described by Carrington and 
McLachlan (1967), Wertz and Bolton (1972), Slichter (1963), 
calculation of the^-factor for simple systems is outlined in 
MQM Chapter 1 1 , so too is the source of the hyperf ine 
structure. See also Memory (1968). Relaxation processes are 
described by Carrington and McLachlan (1967), Poole and 
Farach (1971), and Standley and Vaughan (1968); and a 
collection of important papers has been published by 
Manenkov and Orbach (1966). Detailed theoretical exposition 
will be found in Muus and Atkins (1972). 



electronegativity. The electronegativity of an element in a 
molecule is the measure of its power to attract electrons; the 
greater its electronegativity the greater its drawing power. Two 
definitions of electronegativity are in general use and are due 
to the efforts and insight of Pauling and Mulliken. 

The Pauling electronegativity scale is based on the lowering 
of the calculated energy of a diatomic molecule when ionic 
structures are admitted into its description. Suppose the 
energy of the molecule AB is truly f(AB) but that a calcu- 
lation based only on purely covalent structures yielded 
E eou (AB), the ionic -resonance energy would be E. (AB) — 
F(AB) — £" cov (AB). Pauling found that the square root of 
f ion (AB) could be set proportional to the difference of two 
numbers, one characteristic of the element A and the other of 
B, and that the expression -[f jon (AB}] K = b£-)£l 
was valid for a wide range of combinations. In order to set up 
the scale it is necessary to estimate E (AB): Pauling proposed 
that a reasonable approximation would be the mean of the 
energies of the molecules A 2 and B2 . This was justified by the 
view that they are manifestly nonpolar and that the energies 
of A 2 , AB, and B 2 should form a simple sequence if polar 
structures in AB are omitted. Pauling used both an arithmetic 
mean and a geometric mean to set up his table (see Table 7), 



Having set up the electronegativity scale it is possible to 
deduce a number of molecular properties. First, one can use it 
to estimate bond energies if the A 2 and B 2 bond energies are 
known: this is just a reversal of its definition, but energies 
may be predicted of molecules other than those used to 
construct the scale. Next one may use it to predict the 
tonicity of a bond (the percentage ionic character), and for a 
scale expressed in electronvolts the ionicity is given by the 
expression 161^^ — X^ ' "+" 3' 5 'X^"'XJ^ 2 - From the percent- 
age ionic character it is possible to give a rough estimate of the 
"dipole moment if the bond length is known. 

The other scale, that due to Mulliken, is rather more funda- 
mental, for instead of defining it in terms of unmeasurable 
but estimable quantities he defines it as the mean of the 
"ionization potential and the -electron affinity of the atom, 
and both these quantities are measurable. Thus the Mulliken 
scale is based on x^ 1 = \ t/{A) + EJA)] . A further advantage 
of this scheme is that it is possible to take into account the 
differences in electronegativities of different orbitals on the 
same atom or the dependence of the electronegativity on its 
state of 'hybridization. 

It is not surprising that the two scales are related: at a 
numerical level it is possible to equate X^ ~ X^ 1 with 
2- 78 (x^ ~ X^)-' at a deeper level it is possible to show that to a 
good approximation the value of E. (AB) is determined by 
the energy required to move an electron from A to B, which is 
E (B) — /(A), and the energy to move it from B to A, which is 
E (A) — /(B), and that as in a nonpolar molecule these are 
equal (because neither tendency wins) on rearranging the 
equality in this case of vanishing electronegativity difference 
we find that EJA) + /{A) is equal to E (B) + /(B). Conse- 
quently the difference between these quantities should be 
proportional to the difference of the electronegativities of the 
two atoms. 

Questions. What does the electronegativity of an element 
measure? Why is it a plausible approximation that E (AB) is 
the mean of £(A 2 ) and E(B 2 )? Given that the bond energies of 
the halogens X 2 are 2-1 7 eV for F a , 2-475 eV for Cl 2 , and 
1-971 eV for Br 2 , that the energy of H 2 is 4-476 eV, and that 
of 2 is 5*080 eV, estimate the bond energies of the various 
heteronuclear diatomic molecules that may be formed. 



66 electronic spectra of molecules: a synopsis 

Estimate the ionic character of each molecule. Deduce an 
expression for the dipole moment of a diatomic combination 
in terms of the electronegativities of its components. Why 
should you expect poor agreement with experimental values 
(even if you derive a decent expression)? What is the Mulliken 
definition of electronegativity? In what sense is it superior to 
the Pauling scale? On what grounds would you expect the 
Pauling and the Mulliken scales to be related? What molecular 
energy terms are ignored in this argument? 

Further information. See §5.8 of Coulson (1961) and § 2.1 1 
of Pauling (1960) for a detailed discussion of the role and 
deduction of electronegativities. For Mulliken's analysis see 
Mulliken (1934) and Moffitt (1949b). 



electronic spectra of molecules: a synopsis. A natural 
progression into complexity is from the electronic spectra of 
diatomic molecules into polyatomic molecules, then to the 
consideration of chromophores in complex molecules, and 
then to the spectra of molecules in solids. 

The electronic spectrum of a diatomic molecule contains a 
number of bands, each resulting from a transition from the 
ground electronic state (X) into an excited electronic state 
(A, B, . . .), and the structure of the bands is due to the 
simultaneous excitation of vibration; the intensity of these 
"progressions is determined by the "Franck-Condon principle 
and the other "selection rules. On the vibrational structure 
there is a further structure due to the excitation of rotation, 
and P- and R- "branches are generally visible, and Q-branches 
in some cases. Since the moment of inertia is different in the 
two electronic states the branches tend to a head {see 
Fig. B12on p. 31): when the moment of inertia is larger in the 
upper state than in the lower the R-branch, the branch to high 
frequency, has the head. The vibrational lines get closer to high 
energy because the vibrational levels converge towards the 
dissociation limit: the dissociation energy may be determined 
by observing the dissociation limit or by a careful extrapol- 
ation from lower frequencies (a Birge-Sponer extrapolation is 
often used). In some cases the rotational and vibrational struc- 
ture disappears and then reappears before the dissociation 
limit is reached: this is a manifestation of "predissociation. 



From the spectrum may be determined the 'force-constants, 
dissociation energy, and "anharmonicity of the electronic 
states of the molecule, and the parameters in the molecular 
potential-energy curves. 

The electronic spectrum of a polyatomic molecule is more 
complicated, but the same principles apply and analogous 
information obtained. The return to the ground state is an 
additional subject of interest in polyatomics, especially when 
the molecule loses its energy by "fluorescence or "phosphor- 
escence. In a condensed phase the rotational structure of the 
spectrum is lost, and the vibrational spectrum becomes so 
diffuse that the absorption spectrum is often just a series of 
broad bands. The electronic spectra of transition-metal com- 
plexes is of particular importance, and the transitions may 
often be associated with the "crystal-field splitting of the 
d-electrons: see 'crystal -fie Id theory and °ligand-fteld theory. 
In other cases the transition is a charge transfer transition from 
the metal ions to the ligands: see "colour. In organic molecules 
the absorption can often be associated with the presence of a 
chromophore such as the carbonyl group (tt*-*— n electric dipole 
transition) or a double bond {n*< — k transition): see "colour. 

When a molecule is a part of a crystal lattice other effects 
may be observed; one especially important phenomenon is the 
formation of an "exciton, in which the excitation hops 
through the lattice. 

Further information. See MQM Chapter 10 for a discussion of 
molecular electronic spectra. Books that summarize the 
applications of electronic spectra to the study of a molecular 
structure include Barrow (1962), Whiffen (1972), Dixon 
(1965), King (2964), Jaffe and Orchin (1962), Rao (1967), 
Stern and Timmons (1970), Herzberg (1950, 1966), Murrell 
(1971), andGaydon (1968). Further information about the de- 
tailed topics will be found in the appropriate sections. Gaydon 
(1968) is a very good source of information about how to 
interpret electronic spectra of small molecules and how to 
extract potential-energy curves and dissociation energies. 
Murrell (1971) is concerned with larger, organic molecules, 
and discusses individual chromophores and their interaction in 
solids. Herzberg (1950, 1966) is a mine of detailed information, 
and is a brilliant example of the application of theoretical con- 
cepts to the detailed examination of molecular properties. 



elect ronvolt 



67 



electronvolt. The electronvolt (eV) is the energy acquired by 
an electron when it is accelerated by a potential difference of 
1 V. Since the charge on the electron is —1 -602 X 10~ 19 C the 
energy is equivalent to 1602 X 10" 19 J or 96-49 kJ mol -1 . 
Another useful conversion is 1 eV = 8023 cm" 1 . 



equ ipartltion theorem. The mean energy of each mode of 
motion of a classical system in thermal equilibrium is jnkT, 
where n is the number of quadratic terms (of displacement or 
momentum) needed to specify its energy. As an example, the 
energy of an atom in free space arises from its kinetic energy 
which can be expressed as the sum of the three quadratic terms 
P 2 „l2m,pjl2m, and pi 12m; and so the mean energy of an atom 
in equilibrium at a temperature T is ?kT, and 1 mol of 
monatomic gas will have art internal energy j RT. A molecule 
that can "rotate around three axes will have a mean energy of 
wkT associated with each mode, and so a mean rotational 
energy of jx7". A "harmonic oscillator has an energy 
lp 2 /2m) + -^kx 2 , where x is the displacement from equilibrium, 
and so with each oscillatory mode of a body there will be 
associated a mean energy k T because two quadratic terms 
appear in the expression for its vibrational energy. 

The theorem is a deduction from the Boltzmann dis- 
tribution for the population of energy levels at thermal equi- 
librium, and the assumption that the modes concerned are 
classical. When the modes of motion are quantized the 
theorem fails, and so it is not applicable to the vibration of 
molecules nor to small rotating molecules at low temperatures. 

Questions, 1. State the equipartition theorem. Calculate the 
total mean energy of a diatomic molecule, a linear triatomic 
molecule, a bent triatomic molecule, and methane, on the basis 
that ihe translational and rotational motion are classical and 
the vibrational motion quantized and not excited. Derive an 
expression for the molar "heat capacity of these species. In 
terms of the equipartition theorem discuss the contribution to 
the total energy and the heat capacity of a methyl group in a 
molecule which at high temperatures is able to rotate freely 
about one axis, and at lower temperatures can execute only 
classical torsional motions about the axis. 
2. From the Boltzmann distribution at a temperature 7" show 
that components in the total energy of the form x 2 and p 2 



both contribute an amount -^kT to the mean total energy. 
What would happen to the equipartition theorem in the event 
of quartic terms (x 4 andp 4 ) being involved in the energy? 
What effect has •anharmonieity of the form x and x ? 

Further information. This subject is really within the realm of 
statistical mechanics; therefore see Gasser and Richards' 
Entropy and energy levels (OCS 19) for some of its elementary 
applications. See also §7.5 of Reif (1965) for a helpful dis- 
cussion with applications and §10.3 of Davidson (1962) for a 
deduction and discussion. §332 of Fowler and Guggenheim 
(1965) and §44 of Landau and Lifshitz (19586) are worth 
looking at. See Fowler (1936) and Tolrnan (1938) for 
erudite discussion. 

equivalent orbital. A molecular orbital is one of a set of 
equivalent orbitals if a "symmetry operation applied to the 
molecule transforms the orbital into another member of the 
set. As an example consider one of the C— H bonds in methane, 
CH 4 : this is formed from the overlap of an sp 3 - hybrid orbital 
on the central carbon and a 1s-orbital of the hydrogen. If the 
tetrahedral molecule is rotated into another equivalent 
position another of the sp 3 - 1s bonds is rotated into the 
original position. The four O-bonds constitute a set of equiv- 
alent orbitals. The concept of equivalent orbitals is closely 
related to "hybridization and "localized orbitals. 

Further information. See Appendix 9.2 of MQM and Chapter 8 
of Coulson (1961). Further information can be traced through 
"localized orbitals. 

exchange energy. The Coulombic interaction between two 
electron distributions is repulsive, and its magnitude may be 
calculated by dividing both regions into minute charged 
volumes, calculating the Coulomb energy of interaction 
between each of the charged volume elements, and then 
summing over all the elements to obtain the result which we 
shall write./ (see "Coulomb integral). Unfortunately this gives 
the wrong answer because we have omitted the effect of "spin 
correlation, which can cause electrons of the same spin 
orientation to stick together and those of opposite spin to stay 
apart. Thus if the two electrons have the same spin the true 
average repulsion energy will be less than J, because of the 



68 



exciton 



intrinsic tendency of such electrons to avoid each other. The 
correction we should make changes the average repulsion 
energy to J—K, the correction term K being the exchange 
energy. The name reflects the source of the correction which 
lies in the °Pauli principle and the behaviour of wavefunctions 
when electrons are exchanged. 

Further information. See Chapters 8 and 9 of MQM for a more 
detailed account and examples of the application of the 
concept in atoms and molecules. Good discussions will be 
found in §19.16-19 of Bohm (1951} and §10.89 of Davydov 
(1965). For applications to molecules see Coulson (1961), 
Richards and Horsley (1970), and McGlynn, Vanquickenborne, 
Kinoshita, and Carroll (1970), who give, in Appendix E, a 
guide to the computation of the exchange integrals. 

exciton. Imagine an array of identical molecules in a crystal, 
and let one be excited. This excitation hops from molecule to 
molecule, and slides throughout the crystal until it decays. The 
excitation moves like a particle, and this notion is conveyed by 
the word exciton, which is the name given to this migrating 
excitation. If we think of the excitation as being caused by the 
removal of an electron from one orbital of a molecule (or atom 
or ion) and its elevation to a higher orbital, then the excited 
state of the molecule can be envisaged as the coexistence of an 
electron and a hole. The hopping of this electron-hole pair 
from molecule to molecule is the migration of the exciton. 
When the electron and the hole jump together from molecule 
to molecule as they migrate we have the tight-binding case, 
and the migrating excitation is a Frenkel exciton. It is also 
possible for the electron and hole to be on different molecules, 
but in each others vicinity; this is the weak-binding case and 
the migrating excitation, now spread over several molecules 
(more usually ions), is a Wannier exciton. In molecular solids 
the Frenkel exciton is more common, and that will be our 
interest. 

The formation of an exciton affects the spectrum of a 
species in a solid. This should not be surprising, for the mi- 
gration of an exciton implies that there is an interaction 
between the species composing the crystal (for otherwise the 
excitation onone unit could not move to another), and this 
interaction should affect their energy levels. The strength of 



the interaction governs the rate at which an exciton moves 
through the crystal: strong interaction implies fast migration; 
vanishing interaction implies that the exciton is localized on its 
original molecule, in which case it is an ordinary excitation of 
an 'isolated' molecule, Exciton formation causes lines to shift, 
split, and change in intensity. The reasons for this we shall 
understand when the mode of interaction between the mole- 
cules has been explained. 

An electronic transition in a molecule involves a shift of 
charge (see "transition probability and "electric dipole tran- 
sition). A shifting charge on one molecule exerts a force on a 
neighbour, which can respond with a shift of its charge. This 
process of transition dipole interaction can continue, and the 
excitation can rattle through the crystal (Fig. E8). 

The process can be looked at in a slightly different way by 
considering all possible relative orientations of the transition 
dipole moments of the molecules of the crystal, and then 
seeing what combination a light wave can excite. Let us 
consider a linear array of molecules with transition moments 
perpendicular to the line (the arrows indicate the effective 
direction of motion of the charge when a molecule is excited). 
On the basis of simple electrostatics the energy of the array 
with all transition dipoles parallel is higher than any other 
phasing (Fig. E9), and so a transition to this arrangement of 
dipoles throughout the crystal appears at an energy higher 
than the transition to any other arrangement, and indeed 
higher than the transition in the separated molecules. But it is 
only this arrangement that the incident light is able to excite. 
This is because the wavelength of light is so long in comparison 
with the molecular spacing that its electric field has the same 
phase over a large number of molecules; therefore it stimulates 
a whole domain of transition dipoles to move in phase. This 
means that the presence of the exciton coupling appears in the 
spectrum as a shift of the absorption band to high energy. If 




FIG. E8. An excitation migrating through a crystal as an exciton. 



exciton 



high energy 



Ikjhf stimulates rhis mode 

7r\ /3\ // 





isolated 
molecule 




low energy 



isolated molecule A A exciton 

FIG. E9. Transition dipoies with various phasings. The alternating 
alignment lies lowest, the parallel (the one excited by incident light) 
lies highest. 

the transition dipole moments were along the line of the mole- 
cules we should get the opposite shift; this follows from the 
fact that the in-phase excitation of the transition dipoies gives 
a head-to-tail array, which has a low energy. 

If there is more than one molecule per unit cell it is possible 
to obtain several lines: N molecules per unit cell give N absorp- 
tion lines, or exciton hands. The splitting between the bands is 
the Davydov splitting. To see how it arises consider the case 
when N = 2 and the molecules are arranged as shown in 
Fig. E10. Let the transition dipoies be along the length of the 
molecules. The light field stimulates the in-phase excitation of 
the transition dipoies, but they need be in-phase only in so far 
as neighbouring unit cells are concerned. Within each unit cell 
the transition dipoies may be arrayed as in Fig. E10 a or b, and 
these have different interaction energies. This will appear in 
the spectrum as two bands split by the energy of interaction 
between the transition dipoies within the unit cell. 

Questions. 1. What is implied by the term 'exciton'? What is 
the difference between a Frenkel and a Wannier exciton? 



Would you expect excitons to contribute to the conduction of 
electricity in the medium? How may excitons be recognized 
spectroscopically? Why is it reasonable to consider only the 
in-phase excitation of transition dipoies? What will happen to 
the spectrum as this approximation fails? Assess the wave- 
length at which it might be expected to be seriously in error 
for the 380 nm band of anthracene. 
2, Estimate the interaction energy between two transition 
dipole moments separated by 0.3 nm and each of magnitude 
1 D (3-3 X 10" 30 Cm). Suppose the "oscillator strength of the 
transition in the free molecule is f = Q-2; estimate the exciton 
shift. What is the source of the Davydov splitting? What is the 
polarization of the split bands? 

Further information. A simple account of the formation of 
molecular excitons will be found in Murrell (1971), who also 
discusses hypochromism (the reduction in intensity of absorp- 
tion) and hyperchromism (the increase in intensity). (Further 






solated molecules 
A 



^ 



Davydov splitting 



FIG. E10. Davydov splitting for exciton bands for two molecules per 
unit cell. 



70 



expectation value 



confusion may be encountered with the names hypsochromism , 
which implies the lightening of a colour, as in dyeing, and its 
opposite, bathochromism , the deepening of a colour.) An 
introduction to the mathematical theory of excitons will be 
found in Craig and Walmsley (1968) and Kittel (1971), and 
further development is given by Davydov (1962, 1965). 

expectation value. The expectation value of an observable is 
its mean value for the state of the system, and is the mean 
result of a set of experiments designed to determine the value 
of the observable for that state. Thus the expectation value of 
the energy is the mean energy that would be measured in a set 
of suitable experiments on a collection of identical systems; 
the expectation value of the position is the mean position that 
would be measured, and the expectation value of momentum 
or angular momentum is that mean value. 

If the system is in a pure state with respect to one of the 
observables (in other words the state is an -eigenstate of 
the "operator corresponding to that observable) all identical 
experiments give identical results, and the expectation value is 
the result that would be measured in all the experiments; such 
results are dispersion free. An example would be the determin- 
ation of the energy of an atom or a molecule when it is in a 
definite energy state; it is then in an eigenstate of the energy 
operator for the system, and the expectation value of this 
operator is the energy that would be measured in all 
experiments. 

Quite often the system is not in a pure state but is better 
described by a -superposition of pure states; performing one 
experiment to measure a property of such a system gives a 
result which can be identified as one of the eigenvalues of the 
operator corresponding to the property, and the mean result 
of a set of identical results performed on a set of identical 
systems will be the expectation value for the system in that 
state. An example could be the determination of the linear 
momentum of a system which is described by a "superposition 
of states of different momentum {for example, if the particle 
is described by a -wave packet): if the state with momentum 
kh occurs in the superposition with weight lc. P then the 
experiment will yield the answer kh with a probability \c k I 2 , 
and will yield other values with a corresponding probability. 



The average of all such measurements is the expectation 
value of the linear momentum, written (p>. 

If the state of a system is \p the expectation value <S2> 
of an observable £1 is equal to the integral fdT\p*£l\]j , where 
£2 is the appropriate -operator for the observable, and it is 
supposed that the state \p is -normalized. Therefore we may 
predict the result of a set of experiments by calculating the 
expectation value by evaluating the integral. But in one 
experiment we can expect to observe the value of the integral 
only if the state \p n is an eigenstate of the operator corre- 
sponding to the experiment we are attempting to perform. If it 
is not an eigenstate, the value of the integral tells us only the ' 
mean value of a large number of experiments; this is the only 
information we are able to calculate, and we are unable to 
predict, except as a probability, precisely what result we shall 
get from a single experiment. 

Questions. What is meant by the expression 'expectation value 
of an observable'? In which sense does it determine the result 
of an experiment? What interpretation should be put on the 
expectation value when the state of interest is a mixture (a 
superposition)? What is meant by a pure state? A beam of light 
is constituted from two plane-polarized orthogonal com- 
ponents and the resulting beam can be described as the super- 
position c x \p x + Cyipy, where X and Y refer to the two 
polarization directions. Discuss the result of an experiment 
designed to measure the polarization of the beam. Using the 
expressions in Box 15 on p. 166 calculate the expectation value 
of the kinetic energy and the linear momentum for a -particle 
in a one-dimensional square well, and discuss the results of an 
experiment to confirm the result. 

Further information. See MQM Chapter 4 for a discussion of 
operators, observations, and expectation values. See also 
Feynman, Leighton, and Sands (1963), Kauzmann (1951), 
Pauling and Wilson (1935), and Bohm (1951). For deeper 
discussions of the measurement process see Dirac (1958), 
von Neumann (1955), and Jauch (1968). 

extinction coefficient. The intensity of a beam of light is 
diminished as it passes through an absorbing medium, and 
since the amount each molecule absorbs is proportional to the 



extinction coefficient 



71 



intensity of the light present, the intensity falls exponentially. 
The rate of decay of the intensity is determined by the extinc- 
tion coefficient. The actual intensity follows the Beer-Lambert 
law I = j exp(—cell), where / is the initial intensity, c is the 
molar concentration of the absorbing species, £ the path length 
(which by convention is normally expressed in centimetres), 
and e is the extinction coefficient. The extinction coefficient 
depends on the frequency of the incident light, and is often 
written e[v), where V is the corresponding wave number in 
cm -1 . 

The extinction coefficient is a measure of the "transition 
probability at the appropriate frequency, and is therefore a 
measure of the strength of the "transition dipole and the 



•oscillator strength of a transition, and in turn is related to the 
"Einstein coefficient of stimulated absorption. Molar extinc- 
tion coefficients for a few representative materials are 
recorded in Table 8. The product ce{i>)$. is called the optical 
density of the material at that frequency. 

Further information. See MQM Chapter 10, Appendix 10.2 
for the relation of extinction coefficient and oscillator 
strength; for this relation see -oscillator strength. A discussion 
of the Beer- Lambert law and its applications and limitations 
will be found in Wayne (1970). A compilation of extinction- 
coefficient data and a discussion of its analytical applications 
are given by Mellon (1950). 



F 



Fermi contact interaction. A magnetic nucleus and an 
electron may have a magnetic interaction by virtue of their 
contact: this magnetic interaction is the Fermi contact inter- 
action, and it is a special case of a "hyperf ine interaction. 
But what is the nature of the interaction on 'contact'? It is 
possible to give a variety of explanations, of varying sophisti- 
cation, of the actual mode of interaction; the most pictorial is 
as follows. Consider the magnetic moment of the nucleus as 
arising from the circulation of a current: we replace the 
magnetic moment by an equivalent current loop. At distances 
far from the nucleus the field due to this loop is indistinguish- 
able from the field from a point magnetic dipote; but close to 
the nucleus, or loop, the point-source nature of the field is 
invalid and the magnetic field is characteristic of a circular 




FIG. Ft. An interpretation of the Fermi contact interaction: only an 
electron in an s-orbital can penetrate the nucleus. 



loop of non-vanishing diameter (Fig. F1). Now bring in the 
electron with its spin "magnetic moment. Far from the nucleus 
it experiences a pure dipolar magnetic field, but if it can pen- 
etrate the nucleus it enters a quite different region, where the 
field flows in only one direction. The magnetic interaction 
between this non-dipolar field and the electron's magnetic 
moment is the contact interaction. 

The electron must come into contact with the nucleus if 
the interaction is to operate. Can an electron come into con- 
tact with the nucleus? If it is a p-electron, or a d-electon, etc. 
it cannot, because all such orbital s have a mode at the nucleus. 
An s-orbital has no node at the nucleus, and an electron occu- 
pying one has a non-vanishing probability of being at the 
nucleus. It follows that only s-electrons can show a Fermi 
contact interaction. Since s-orbitals are spherically sym- 
metrical it also follows that the interaction should show no 
directional characteristics, and indeed it is found to be 
isotropic. 

How strong is the interaction? A measure of its strength is 
the extra magnetic field that an s-electron experiences by 
virtue of its interaction with the nucleus. For the Is-orbital in 
the hydrogen atom the contact interaction (of energy 1420 
MHz) is equivalent to a magnetic field of 508 G acting on the 
electron; for an electron in a 2s-orbital of fluorine the field is 
as strong as 17 kG. The inner electrons of heavier elements can 
possess tremendously large interactions, amounting to 
megagauss. Some representative values are listed in Table 9. 

The Fermi contact interaction plays an important role in 
^electron spin resonance spectra, because the extra local field 
appears in the spectrum as -hyperfine structure. The contact 



72 



fermion 



73 



interaction, being isotropic, does not vanish in fluid media. It 
also plays an important role in "nuclear magnetic resonance 
because it is a contribution to one of the mechanisms of 
"spin-spin coupling. 

Questions. 1, What is the Fermi contact interaction? In what 
sense does it depend on the non -vanishing si2e of the nucleus? 
Why is the contact interaction characteristic of an s-electron? 
Why is the interaction isotropic? Where does the interaction 
play an important role? 

2. The energy of the Fermi contact interaction is given by the 
expression — {2fi Q /3) [y J e l^(0)l 2 ] 1. 1, where 7 N and y e are 
the rnagnetogyric ratios of the nucleus and electron respect- 
ively (see "magnetic moment), I and s their spins, |U the 
vacuum permeability, and li^O)! 2 the probability that the 
electron is at the nucleus. Show that this interaction has the 
form suggestive of a magnetic field (Uq/S) y N \\p{0}\ 2 I arising 
from the nucleus and affecting the electron magnetic moment, 
and from the expression for the hydrogen atomic orbitals given 
in Table 15 on p. 275 compute the magnetic field (actually the 
magnetic induction) experienced by an electron in a Is- and 
2s-orbital of hydrogen. Repeat the calculation for a selection 
of °Slater-type atomic orbitals for the first-row elements. In 
each case replace I by its maximum projection m h. 

3. An electron in an orbital centred on a neighbouring nucleus 
may have a Fermi contact interaction whatever the nature of 
the orbital, provided the orbital has a non-vanishing amplitude 
at the nucleus of interest (thus a 2p-orbital on B may have a 
non-zero amplitude at the nucleus of its neighbour A). The 
strength of the interaction is determined by the expression in 
Question 2 but with f^(0)P replaced by l^(R)l 2 , where R is 
the position of the nucleus relative to the nucleus on which 
the orbital is centred. Calculate the strength of the interaction 
for a proton as it is brought towards the position of another 
proton surrounded by a 1 s-electron. Ignore the distortion that 
the second proton induces. 

Further information. See MOM Chapter 11 for a further 
discussion of the interaction. An account of the role the inter- 
action plays will be found in McLauchlan's Magnetic resonance 
(OCS 1) and in Atkins and Symons (1967), Carrington and 
McLachlan (1967), Slichter (1963), and Abragam and Bleaney 
(1970). Magnitudes of the interaction for a variety of nuclei 



are given on p. 21 in Atkins and Symons (1967). Calculations 
involving the contact interaction are described by Memory 
(1968) and Freeman and Frankel (1967). The derivation of 
the contact interaction is described in a simple manner in 
MQM Chapter 1 1, and performed in §4.5 of Slichter (1963); a 
derivation from the Dirac equation is given in Griffith (1964) 
and Bethe and Salpetre (1957). 

fermiOfl. A fermion is a particle possessing an intrinsic "spin 
angular momentum characterized by a half -integral spin 
quantum number (s or /). Examples include the electron 
(s = h, the proton (/ = j), the neutron (/ = j), the neutrino 
(s = |), 3S C1 nucleus (/=§), 13 C{l = j), and 17 (/ = §). 
Ferrnions obey the -Paul! exclusion principle, and so no more 
than one can occupy any single quantum state. This has a 
profound influence on their behaviour, and distinguishes them 
sharply from -bosons, any number of which may enter a given 
state. The "Pauli principle requires a wavefunction to be 
-anti symmetrica I under the interchange of any pair of identical 
ferrnions. 

Further information. See -spin and the °Pauli principle for 
further discussion. The occupation restriction of ferrnions is 
taken into account by Fermi-Dirac statistics when large 
collections are under consideration: these are described by 
Gasser and Richards in Entropy and energy levels (OCS 19), in 
Chapter 6 of Davidson (1962), and in §9.7 of Reif (1965). 

fine Structure. The fine structure in an atomic spectrum is 
the splitting between different levels (different values of J of a 
particular "term). In atomic sodium the energy of 2 Pj/2 differs 
from the energy of 2 P 3 /2, and so the emissions 2 Pi /7 — * 2 Si/2 
and z P 3/2 — ► z Si /2 occur at slightly different frequencies (and 
give rise to the two closely-spaced yellow D -lines of the sodium 
spectrum); this is an appearance of fine structure. 

Fine structure is a manifestation of spin-orbit coupling and 
is best introduced by considering a one-electron atom. Suppose 
that the electron in the atom has an orbital angular momentum 
1: by virtue of its charge it also has a "magnetic moment, 
which by Ampere's law may be considered to be a dipole at 
the centre of the orbit. This magnetic moment gives rise to a 
magnetic field which interacts with the spin magnetic moment 



74 



f i ne-structu re co nsta nt 



of the electron, and the interaction energy depends on the 
relative orientation of the electron's spin and orbital magnetic 
moments. This implies that the energy also depends on the 
relative orientations of the angular momenta 1 and s (see 
Fig. F2). The low-energy orientation corresponds to the 
opposition of ji. and /i„, and therefore it also corresponds to 
the opposition of 1 and s; consequently the lower value of the 
total angular momentum j (j = 1 + s) corresponds to the lower 
energy. 




FIG. F2. The magnetic interaction resulting in fine structure. A high 
total angutar momentum corresponds to a parallel alignment of 
moments, and therefore to a high energy. 

We see that different values of/ (and of J in many-electron 
atoms) correspond to different energies by virtue of the 
magnetic spin-orbit interaction. This is the basis of the third 
°Hund rule: when an electron shell is less than half full, low 
values of J have lower energy than high values of J (the 
opposite is true when the shell is more than half full). "Terms 
in which high-/ levels have higher energies than low-/ levels 
are called regular; when low-/ levels lie highest the term is 
inverted. Since a term with several/ values (for example, a 2 P 
term which has the two levels 2 P[/2 and 2 P 3 /2) is called a 
"multiplet term, we arrive at the names regular multiplet and 
inverted multiplet. 



Within a given multiplet the spacing of the levels obeys the 
Lande interval rule which states that the energy interval be- 
tween pairs of adjacent levels is in the ratio of the /-values of 
the upper level of each pair. Thus the ratio of the 3 Pj— Pj 
interval to the 3 P, - 3 P interval is 2:1. 

As the spin-orbit coupling increases with atomic number we 
both expect and observe the fine structure to be very import- 
ant in heavy atoms. 

Further information. The fine structure is discussed in more 
detail in MQM Chapter 8. See especially §111 A. 5 of Kuhn 
(1962), where the mathematics is developed in a simple way; 
atoms more complex than hydrogen conform to the same 
principles, and are discussed by the same author in §lll D.3 
and Chapter V. See especially Chapter 2 of Herzberg (1940). 
Discussions of the multiplet structure and fine structure of 
atoms is also described in detail by Woodgate (1970), Condon 
and Shortley (1963), Griffith (1964), and Shore and Menzel 
(1968). 

fine-Structure constant. The fine-structure constant is a 

measure of the strength of interaction between a charged 
particle and the electromagnetic field: it is given by the 
expression a = e~/4Tre hc; it is dimensionless and has the 
numerical value 1/13703602, or approximately 1/137. The 
smallness of this number is of great importance, for it 
determines the size of atoms and the stability of matter. If 
<x were much larger the distinction between matter and radi- 
ation would be much less clear; if it were much smaller, matter 
would have virtually no electromagnetic interactions. 

These considerations can be elucidated by considering how 
the fine-structure constant (that is, the strength of the coupling 
between charged matter and the electromagnetic field) deter- 
mines the size of atoms and the magnitude of some of their 
properties. Consider an atom of radius r in which the potential 
energy of the electron is of the order of Ze 2 /47re r: it is tempt- 
ing to express this in terms of a, and falling into temptation we 
obtain Zcihclr. The kinetic energy of the atom is of the order 
of p 2 /2m ; if we use the -uncertainty principle to assess the 
order of magnitude of this term on the basis of an electron 
being confined to a region of radius r we may use p~hlr. This 
enables the total energy to be expressed in terms of the 



fluorescence 



75 



parameter r, roughly the size of the atom, and to seek r by 
minimizing the total energy (finding the value of r for which 
df/dr = 0). The answer we get is r-X /Za, where X is the 
"Compton wavelength of the electron (A_ = h/cm ~ 
2-4 X 10 m). This shows that the size of an atom is roughly 
this characteristic 'size' of an electron times 137. It is 
interesting to note that the "Bohr radius can be expressed as 
X_/47Ta, or as r Id 2 , where r is the classical radius of the 
electron. If the interaction strength were much less the atom 
would be much larger, and if a were much larger, and com- 
parable to the analogous coupling constant for the nucleon 
interaction (the 'strong interaction' as opposed to the electro- 
magnetic interaction), then atoms wouid be of roughly the 
same size as their nuclei. The minimum energy can be found 
by substituting the size of the atom back into the energy 
expression, and rather pleasingly we i'md - IrZ 2 a 1 (m c 2 ): this 
is pleasing because m c is the relativistic expression for the 
rest energy of a particle of mass m , and so the energy of an 
atom is of the order ofZ 2 /137 7 of this value. This is a small 
proportion when Z is small, but it may approach unity as Z 
gets large (in heavy atoms). This implies that ordinary non- 
relativistic quantum mechanics is good for light atoms, but 
fails progressively through the periodic table. Another amusing 
deduction is the order of magnitude of the velocity of an 
electron in an atom or molecule: simple consideration of the 
energy expression (see Questions) gives the answer v~Zac: 
since a is so small, electrons move at non-relativistic velocities 
except in the case of heavy atoms. Another place where the 
fine-structure constant enters is in the magnitude of the "spin- 
orbit coupling which determines the "fine structure of spectra: 
in hydrogen-like atoms the strength of this interaction is pro- 
portional toa 2 ff, where R is the Rydberg constant, and it is 
from this relation that a gets its name. 

The stability of atoms is determined at several levels by the 
small size of a. One important role played is the way that the 
"transition probability for -electric dipole radiation (essentially 
a mechanism whereby an atom emits or absorbs a photon) 
depends on a: an analysis of the situation shows that an elec- 
tron has to oscillate about 1/a(aZ) 2 times before it is virtually 
certain to emit a photon: this accounts for the moderate 
stability of the excited states of atoms and molecules. The 
probability that two photons are thrown off by an excited 



atom is of the order of the square of the probability that one 
will be emitted, and so this process is of very low probability 
and is indeed only rarely observed. But matter can also decay 
into radiation: matter can be annihilated and appear as 
electromagnetic energy. The probability of an electron dis- 
appearing in this fashion is proportional to its strength of 
coupling to the electromagnetic fields, and varies as a. Put 
another way, an electron spends a of its time as electro- 
magnetic radiation. Fortunately a is small; if it were closer to 
unity, matter and radiation would be indistinguishable. 

Questions. 1. Of what is the fine-structure constant a measure? 
What would be the consequences of it being zero? Evaluate 
its magnitude from the values of the fundamental constants, 
and confirm that it is dimension less. What features of atomic 
and molecular structure does it determine? Express the spin- 
orbit coupling parameter, the °Bohr radius, the -Rydberg 
constant, and the "Einstein coefficients in terms of a, and 
attempt to interpret the form of the expressions. Carry through 
the calculation of the hydrogen-atom energy as described in 
the text: find the lowest energy as a function of atomic size r, 
the size of the atom in this case, and the velocity of the elec- 
tron (from the kinetic energy). 

Further information. A very good and quite simple account 
of the way that the fine-structure constant determines the 
magnitude of atomic properties is given in Chapter 1 of 
Thirring (1958). 

fluorescence. Light when absorbed by a molecule may be 
degraded into thermal motion or it may be re-irradiated. Light 
emitted from an excited molecule is fluorescence or phos- 
phorescence. The two types of radiation may be distinguished 
by the mechanism that generates them and, more ambiguously, 
by the general observation that fluorescence ceases as soon as 
the exciting source is removed, whereas phosphorescence may 
persist. See "phosphorescence for an account of its mechanism. 
The mechanism of fluorescence is illustrated in Fig. F3. The 
incident light excites a ground-state molecule into a state which 
we shall label S t ; assuming the ground state is a -singlet state 
(all electrons are normally paired in molecules) and we are 
interested in a strong initial absorption, the upper state is also 
a singlet. The molecule is also excited vibrational ly during the 




FIG. F3. The processes leading to fluorescence, and the 'mirror* image' 
relation of the absorption and Fluorescence spectra. Note the shift of 
the latter to longer wavelengths. 



transition (see -Franck-Condon principle). Collisions with the 
surrounding medium, which may be a gas, a solvent, or a solid 

lattice, induce vibrational transitions because the surrounding 
molecules may be able to carry away the moderately small 
amounts of vibrational energy of the molecule, and so to lower 
it down the ladder of vibrational states in the upper electronic 
state. When the molecule has reached its lowest vibrational 
state two things may occur. One is that the solvent may be 
able to carry away the electronic energy and so deactivate the 
molecule: this the solvent may do if it has an energy level that 
matches the energy of the excited molecule, for there may 
then be a 'resonant transfer of radiation to the solvent, which 
then fritters away its excitation into thermal motion. An 
alternative mode of decay, and the one that concerns us o1 
present, is the fluorescent decay of the excited electronic state: 
the molecule deactivates by emitting a photon and falling back 
into the lower electronic state. This emitted light is the 
fluorescence. 

Whether or not fluorescence appears depends on compe- 
tition between the radiattonless deactivation, involving energy 
transfer to the surrounding medium, and the radiative emission. 



In a gas a molecule receives about 10 10 collisions per second, 
and in a liquid the continuous jostling by the solvent amounts 
to about 10 12 — 10 13 collisions per second. Observed lifetimes 
of fluorescent radiation are of the order of 10~ 8 s, and so in a 
gas we have to consider the effectiveness of about 100 
collisions, and in a solvent 10 4 — 10 s collisions. If the 
collisions are strong in the sense that they are effective in 
absorbing energy then the molecule will be deactivated by the 
radiationless processes, especially when the molecules are in a 
liquid. If the collisions are unable to extract much energy, 
then even in a solvent they may be able to lower the elec- 
tronically excited molecule down only the vibrational ladder, 
and they will be unable to extract an electronic excitation 
energy. In this case the radiation decay dominates and the 
molecule fluoresces. 

Two characteristics of the fluorescence should be noted. 
The first is that the fluorescence should appear at lower 
frequency than the incident light. This may be seen im- 
mediately from Fig. F3, which shows that the energy of the 
emitted photon differs from that absorbed by the amount of 
vibrational energy lost to the surrounding medium. Therefore 
we might expect objects irradiated with blue or ultraviolet 
light to fluoresce more in the red. Bright-red fluorescent 
clothing is a part of the modern scene, and a manifestation of 
this effect. The second point is that there may be vibrational 
structure in the spectrum of the fluorescent light: this 
vibrational structure is a "progression formed by the decay of 
the ground vibrational state of the upper electronic level into 
different vibrational levels of the lower electronic level. Its 
study can provide information about the "force -constant of 
the molecule in its ground state, and this is in contrast to 
normal electronic spectra which provide information about the 
stiffness of the bonds in the upper electronic level. It follows 
that the absorption spectrum and the fluorescence spectrum of 
a molecule should resemble each other: this is normally ex- 
pressed by saying that one is the 'mirror image' of the other 
{see Fig. F3); but that description should not be taken 
literally, because the vibrational splittings and intensities are 
not quite the same. 

A number of details may be added to this basic description. 
The first is that the initial absorption might not take place to 
the lowest excited singlet state of the molecule. In this case an 



fluorescence 



77 



internal conversion occurs in which the higher singlets S 2 , S 3 , 
etc. are induced, by a collision, to make a radiation less trans- 
ition into the lowest excited singtet S] , which then fluoresces 
(see °Auger effect). A famous rule due to Kasha reflects this 
effect: the fluorescent level is the lowest level of that multi- 
plicity (for example, the lowest excited singlet level). The 
intensity of fluorescence depends strongly on the physical 
state of the sample because of the deactivating collisions in 
competition with the fluorescence; pure, undiluted liquids 
generally have a very low fluorescent efficiency because the 
excitation may hop from one molecule to an identical 
neighbour by a resonant process (see °exriton for the 
analogous effect in solids). Conversely, it is possible to en- 
hance fluorescence by having present a molecule that can 
absorb the incident light and then transfer it (by a matching 
of energy levels, and a collision) into a molecule that may then 
fluoresce: this is sensitized fluorescence and is made use of in 
some kinds of "laser. Another term often encountered is 
resonance fluorescence; this signifies that the fluorescent 
radiation has the same frequency as the incident light; when 
this is so the fluorescence may be brighter because the 
transition is stimulated. Light of exactly the same frequency 
is rare in fluorescing molecules because the presence of the 
solvent slightly shifts energies, and so the (0-0) vibrational 
upwards transition {that is the transition from the lowest 
vibrational level of one state to the lowest of the other) might 
differ in energy from the (0—0) downwards transition because 
the solvent may solvate the upper state differently before the 
fluorescence occurs. Fluorescence is generally extinguished as 
soon as the incident illumination ceases: this is because al) the 
transitions of interest are allowed, and therefore occur very 
quickly. Neverthefess, there is the phenomenon of delayed 
fluorescence (not to be confused with phosphorescence) 
which may persist for several milliseconds. The mechanism 
for this depends upon the excitation of a molecule from the 
ground state S into the singlet S l( and then this molecule 
migrating to another molecule to which it sticks by sharing its 
excitation. Thus we have the reaction S + S* — > (SoS^*. 
This excimer (if the two molecules are the same) or exciplex 
(if they are different) then falls apart, after a short life, with 
the emission of fluorescent radiation, 



Questions. What are the characteristics of fluorescent radiation? 
How may it be distinguished from phosphorescent radiation? 
What electronic processes are responsible for fluorescence? 
What other alternative paths of energy degradation are open to 
atoms and molecules in gases and solutions? What properties of 
the solvent determine the rate of non-radiative decay? What 
differences would you expect in the fluorescent behaviour of 
a molecule dissolved first in a strongly interacting solvent with 
high-frequency bending and stretching vibrations, such as water, 
and then in a weakly interacting solvent with flabby bonds, for 
example selenium oxychloride? What is a fluorescent spec- 
trum of a molecule? How does this information complement 
the absorption spectrum? In what sense is the fluorescent spec- 
trum a mirror-image of the absorption spectrum? What is the 
role of the °Franck-Condon principle in determining the 
structure of the fluorescence spectrum? What is meant by the 
term 'internal conversion", and what is its significance in the 
study of fluorescence? In what sense is fluorescence an "Auger 
process? Why is fluorescence largely quenched in the pure 
liquid? How may fluorescence be enhanced? What is resonance 
fluorescence? In what sense does the operation of some types 
of *laser depend on the mechanisms described here? Why does 
the fluorescence spectrum show a shift from the position of 
the absorption spectrum, even allowing for the mirror-image 
symmetry? To answer this, consider a molecule surrounded by 
a polar solvent; and then excited by a transition that changes 
the molecule's polarity (for example, a jr*«— n -electric dipole 
transition). Let the polar solvent relax about the excited state, 
and consider the energy of the emissive transition. What is 
meant by the terms excimer and exciplex, and what is a conse- 
quence of their formation? How may delayed fluorescence be 
distinguished from "phosphorescence? 

Further information. See MQM Chapter 10 for more discussion. 
A simple account of some of the relaxation processes described 
has been given by Heller (1967) in connexion with liquid lasers. 
See also Haught (1968). A good account of fluorescence and 
related processes has been given by Bowen (1946), Wayne 
( 1 970), and Calvert and Pitts ( 1 966) . The generation of I ight 
in chemical reactions {chemiluminescence) is a phenomenon 
related to fluorescence, the difference being that the excited 



78 



force -constant 



state of the emitting molecule is formed as the product of a 
chemical reaction. This subject is described by Wayne (1970). 
Energy-transfer processes are at the root of the fluorescence 
efficiency; therefore see Levine and Bernstein (1974). 

force-constant. The force-constant k is the constant of 
proportionality between the restoring force and the dis- 
placement x of a simple harmonic oscillator: force = ~kx. 
Large force -constants imply stiff systems (strong restoring 
forces even for small deviations from equilibrium). The 
frequency of a classical simple harmonic oscillator is related to 
k by the expression o> = {klm) v ' , where m is the mass of the 
oscillating system (with k expressed in newtons per metre and 
mass in kilograms, the frequency will be in radians per second; 
to get hertz divide by 2ir). The lesson taught by this expression 
is that the frequency of oscillation is determined by the mass 
as weli as the force-constant, for the heavier the mass the less 
effective will be the restoring force. In the quantum-mechanical 
treatment of the "harmonic oscillator the energies are given in 
terms of the fundamental frequency loq calculated in the same 
way as the classical case: the energy of the nth quantum level 
is (n + jJ^^- 

The importance of the force -constant in quantum theory is 
that it is a measure of the stiffness of bonds between atoms, 
and therefore governs (together with the atomic masses) the 
"vibrational frequencies of molecules. 

Table 10 lists typical values of force-constants for some 
molecules and the corresponding frequencies and quantum 
energy-level separations. 

Questions, 1. What is the force-constant? What is the physical 
significance of a large force -constant? Would you expect the 
force-constant for the C— C bond in diamond to be less than 
that for the Pb— Pb bond in metaliic lead? A mass of I kg 
hangs from a spring with force-constant 1 N m" 1 : what is its 
natural frequency in radians per second and in hertz? To what 
should the mass be changed in order to oscillate at iHz? What 
force-constant would be needed if the mass were that of a 
proton and the frequency that typical of a molecular bond 
(~-10 l4 Hz)? What wave number (cm -1 ) does this correspond to 
in the infrared absorption spectrum? Where is the absorption 
shifted to on deuteration? 



2. By a Taylor expansion of the bond energy about its 
equilibrium value, show that the force-constant is proportional 
to the curvature of the molecular potential -energy curve at the 
equilibrium separation. 

Further information. See MQM Chapter 10 for more infor- 
mation about molecular vibrations. A good, simple, but 
detailed account is given by Woodward (1972), who reveals 
how to determine k from vibrational data, and a standard work 
is that of Wilson, Decius, and Cross (1955), See also Whiffen 
(1972), King (1964), and Gans (1971). A complication of 
molecular vibrational data has been made by Adams (1967, 
1971). 

Franck-Condon principle. The Franck-Condon principle 
governs the intensity of spectral transitions between the 
vibrational levels of different electronic states of molecules. 
By recognizing the great difference in mass between the 
nuclear framework and the electron being excited, it states that 
the electronic transition occurs so rapidly that during it the 
nuclei are static. A vertical transition occurs, which begins with 
the nuclei in some arrangement in the lower electronic state 
and ends with them in the same arrangement in the upper 
electronic state. But as the molecular energy curves might be 
displaced (see Fig. F4) this nuclear arrangement might corre- 
spond to a highly compressed or stretched state of the excited 
molecule, and so the molecule immediately starts to vibrate. It 
follows that a vibrational excitation of the molecule generally 
accompanies an electronic transition. 

Which of the vibrational levels is most populated by the 
transition is governed by the relative positions of the upper 
and lower energy curves: if the curves were of the same shape 
and one lay directly over the other (Fig. F4a), the transition 
would be from the ground vibrational level of one to the 
ground vibrational level of the other, and so the electronic 
transition would occur without vibrational excitation. In 
general, we may envisage the transition as occuring from the 
most probable conformation of the ground state— which is the 
static, equilibrium arrangement of the nuclei. The electronic 
transition occurs, and during it the nuclei do not change their 
arrangement. At the completion of the electronic transition 
the nuclei are static, but in a new force-field because of the 



Franck-Condon principle 



79 




FIG. F4. The classical basis of the 
Franck-Condon principle. The bob 
remains static during the excitation, 
and the amount of vibrational 
energy simultaneously excited 
depends on the relative disposition 
of the potential-energy curves. 



new electronic distribution. They therefore begin to move, and 
swing harmonically away from and back to their initial 
arrangement (Fig, F4b and c). It follows that the original 
arrangement is a turning point of the new motion, and that 
vibrational energy is stored by the molecule. A line drawn 
vertically from the initial ground state intersects the upper 
potential -energy curve at the point which will be the turning 
point in the excited state, and which shows how much energy 
is absorbed in the transition. (Remember that the energy of a 
"harmonic oscillation is constant: what potential energy it 
loses as the spring decompresses is turned into kinetic energy 
which is used to recompress the spring. Therefore the potential 
energy at the turning point, E in Fig. F4, determines the 
energy at all displacements for that mode of oscillation.) 
The quantum-mechanical basis of the principle is the 
"overlap between the "vibrational wavefunctions of the two 
electronic states: transitions occur most strongly between 
vibrational states that overlap most, because two states that 
overlap strongly have similar characteristics. The ground 
vibrational wavefunction is a bell-shaped curve with its 
maximum at the equilibrium nuclear conformation (Fig. F5). 
Many of the vibrational wavefunctions of the excited elec- 
tronic states overlap this function, but the greatest overlap 
occurs with functions that peak in the same region of space. If 



the energy curves are displaced, the peaks of importance are 
those that occur at the edge of the potential well (see Fig. F5), 
and so the vibration excited will be that predicted by the 
simple device of drawing a vertical transition from the equi- 
librum separation in the ground state (the most probable con- 
formation, and where the vibrational wavefunction peaks 
strongly) to the point where it intersects the edge of the upper 
potential curve. In the vicinity of this intersection the vi- 
brational wavefunctions have moderately strong amplitudes, 
and so overlap most strongly with the ground vibrational 
state. The observed distribution of vibrational intensities then 
reflects the different overlaps between the ground and excited 
vibrational state wavefunctions. 

Questions. 1. State the Franck-Condon principle. On what 
does its validity depend? What is the 'classical' explanation of 
the principle? What is the quantum-mechanical explanation, 
and how is the classical explanation related to it? What is a 
vertical transition? Construct a diagram similar to that in 
Fig. F5 in which the Franck-Condon principle is applied to 
determine the intensity distribution of "fluorescent transitions 
(from the ground vibrational level of the upper electronic 
state to various levels of the lower electronic state). 
2. Demonstrate the quantum-mechanical basis of the principle 
by considering the "transition dipole moment between the 



80 



Fra nek -Hertz experiment 




FIG. F5. The quantum basis of the Franck-Condon principle. The 
strongest transition occurs to the state with which the lowest vi- 
brational level of the lower state has the greatest overlap; this is 
shown shaded. The resulting spectrum is shown on the left. 



two states. Proceed by supposing that the state of the mole- 
cule can be written as the product of the vibrational state and 
the electronic state, and then think about the transition 
moment <e*, v* Idle, v). Show that this may be approximated 
by an expression of the form (e* ldle> {V* If), and recognize 
the presence of the overlap integral between the vibrational 
levels of the electronic states. Calculate the -overlap integral 
between the ground states of two "harmonic oscillators whose 
equilibrium conformations are displaced by a distance R, and 
plot the (0-0) transition intensity (the transition V = to 
V = 0) as a function of R. 

Further information. See MQM Chapter 10 for a further 
account, and a deduction of the principle. See also §6.16 of 
King (1964), Whiffen (1972), and Barrow (1962). A thorough 
discussion of the basis of the principle is given in §IV,4 of 



Herzberg (1950) and developed in even more detail in 
Herzberg (1966). The original formulation of the principle 
was by Franck (1925), and this was turned into mathematics 
by Condon (1928). A useful summary of modern work in- 
volving the principle has been given in Nicholls (1969). 

Franck-Hertz experiment. In the Franck-Hertz experiment 
a beam of electrons was passed through a gas at low pressure 
and the current arriving was monitored; in later experiments 
the energy of the electron arriving was also monitored. As the 
energy of the incident electrons was increased it was found 
that the current arriving at a collector dipped sharply when the 
incident energy was equal to some excitation energy of the 
atom or molecule. It was observed that the sample simul- 
taneously emitted light of a frequency corresponding to the 
energy of the incident beam. This can be rationalized in terms 
of the quantization of energy, for the electrons are able to 
donate their kinetic energy to an atom only if the atom can be 
excited by that amount; therefore the current will dip each 
time the energy can be imparted to the quantized system. This 
is confirmed by the observation of the emitted radiation at the 
corresponding frequency [hv= Af). 

One of the important features of the experiment is that it 
is an illustration that energy is quantized even when it is 
imparted from mechanical motion, as opposed to electro- 
magnetic radiation. 

Further information. See Chapter VI of White (1 935) for a useful 
introduction. More discursive accounts are given in §2.15 of 
Bohm (1951) and p. 85 of Jammer (1966). The original papers 
are those of Franck and Hertz (1914, 1916, 1919). 

free valence. An atom may be linked to its neighbours by 
bonds of various order ( "bond order); the free valence of the 
atom is the difference between its maximum possible total 
bond order and the actual total bond order, and it therefore 
reflects the lack of saturation of the valence requirements of 
that atom. If triple bonds are discounted, for carbon the 
maximum total bond order is 473 (or 3 + /3). As an example, 
in the 'benzene molecule each carbon atom has a bond of 
order 1 to its hydrogen atom, two a-bonds to the neighbouring 
carbons, and two ?r-bonds of order \; therefore the total bond 



free valence 



81 



order of each carbon is 4-33 and the free valence is 0-40, In 
butadiene the C— C bond orders are 1-89, 1-45, and 1-89 along 
the chain, and so the free valences are 0'84 for the two outer- 
most atoms and G-39 for each of the inner pair. 

The magnitude of the free valence is a quantitative measure 
of Thiele's early theory of partial valence, which allows some 
predictions to be made about the relative reactivity of atoms 
in conjugated chains. Thiele supposed that because only one 
bond was sufficient to hold two carbon atoms together each 
carbon atom in the chain had a partial valence which could 
be used for reacting: the remaining part of the double bond 
was superfluous. Nevertheless he took the view that the two 
central partial valences got tangled up, leaving only the outer 
atoms with available valencies. Now, however, we can interpret 
the reactivity in terms of the different free valencies on the 



atoms. Free valencies also occur at the surfaces of metals, and 
the chemical consequences of this include the immensely 
important role of catalysts. 



Further information. See §9.1 1 of Coulson (1961), and §2.9 
and several other sections in Streitweiser (1961). See also 
Daudel, Lefebvre, and Moser (1959), and Pullman and 
Pullman (1958). For lists of bond orders see Coulson and 
Streitweiser (196S). For a discussion of the reactivity of 
molecules in terms of the concepts of free valence, and other 
quantities, see Chapter 1 1 of Streitweiser (1961). For a dis- 
cussion of catalytic activity see Bond's Heterogeneous catalysis: 
principles and applications (OCS 18). Why 4-73? See 
Moffitt (1949a). 



G 



g-value. The "magnetic moment of an electron that arises 
from its -orbital angular momentum is 7 1, where 7 is the 
magnetogyric ratio; but the magnetic moment of an electron 
due to its spin angular momentum is 07 s, where g is an 
additional 'anomalous' factor to which experiment ascribes 
the value 2-0023 (often approximated to 2). Like most 
'anomalous' quantities an explanation can be found in a 
deeper theory, and indeed it should not be surprising to find 
an extra factor of 2 appearing in connexion with spin (which, 
after alt, has no classical analogue, and involves />a/f-integral 
quantum numbers). The deeper theory required is that of 
"Dirac: his relativistic quantum mechanics leads naturally to 
the deduction that g = 2; but the theory is too strict, for it 
requires g to be equal to the integer 2. This shows the Dirac 
theory to be incomplete: the extra 0/0023 required for 
the observed value can be found from the even deeper 
theory of "quantum electrodynamics. In this theory the 
electron is continuously buffetted by stray electromagnetic 
fields which are always present, even in an ideal vacuum: these 
fields affect the spin of the electron in such a way, that the 
magnetic moment is increased from its Dirac value. The cal- 
culation of the 0-value of the free electron is one of the 
triumphs of quantum electrodynamics. 

The LandS g- factor is closely related to thej-value we have 
just described; indeed, they are identical in the limit of vanish- 
ing orbital angular momentum. The La nd e j-f actor determines 
the effective magnetic moment of an electron or atom possess- 
ing both spin and orbital angular momenta, which are com- 
bined together to give a total angular momentum J. For an 



atom described by the quantum numbers S, L, and J the Lande 
tf-f actor is ffj= 1 + ^ {[JU + 1} + S(S+ 1)-£(L+ 1)]/ 
J{J + 1 )}. When L = we obtain g J = 1 2 because J can then 
equal only 5; and when S = we obtain g, = 1, which is the 
normal value for a spinless system. In terms of the Lande 
factor the magnetic moment of a system with angular 
momentum J may be written g,y J. 

The peculiar form of gj arises from the anomalous p-value as 
follows. The vector-coupling picture of a system with spin 
momentum S and orbital momentum L coupled to give a 
resultant J is shown in Fig. G1. L and S both "precess around 
J, and J processes around some other axisz. Antiparallel to 
both L and S we may draw vectors representing the corre- 
sponding magnetic moments, but the spin magnetic moment 
must be drawn twice as long in proportion to the orbital 
moment because of the factor g = 2. The resultant of ji. 
and « s , denoted ji'j, does not lie along the direction of J (it 
would if g = 1, as shown in Fig. Gib}, but will precess about 
it because of the precession of L and S about J. Only the 
component along J does not average to zero during this 
precession, and so the effective magnetic moment of J, which 
we write fy, depends on this component, and therefore on the 
value of L and 5. When L> S or S> L,^j lies almost com- 
pletely along J and g, is approximately t or 2 respectively, 
because either the orbital or the spin moment is dominant; but 
when L and 5 are similar, the effective part of ju' may be much 
smaller than its true magnitude. Because J processes around the 
z-direction so too does ju,; therefore if the projection of J on 
z is J z , the projection of ju, on z hg.y J , where g, is some 



82 



g-value 



83 





FIG. G1. In (a) is illustrated the 
source of the Lande'fl-factor; 
[b] is a hypothetical situation in 
which g - 1. 



factor to be determined from the geometry of the situation 
depicted in Fig. Gla. It turns out that this factor is the Lande 
factor already quoted. 

The g-value in 'electron spin resonance is related to both 
these ^-values. In this technique, which is described in more 
detail in the appropriate section, an oscillating field of fre- 
quency V is brought into "resonance with a spin system held in 
a magnetic field B. The energy of the oscillating field is bv, 
and the energy separation of the electron spin levels (that is, 
the energy separation between m — + -^ar\6fn = — 5") is equal 
foffjUgS: the resonance condition is therefore hU — gii^B. The 
g-f actor in this expression would be the free-electron value <7 if 
the electrons under consideration were free, or the Lande'ffj- 
value if they were bound to an atom. In fact the electron investi- 
gated is normally part of a molecular system, and so neither situ- 
ation holds. To a very good approximation (except in some 
transition-metal ions) an electron in a molecule possesses no 
orbital angular momentum; therefore we can expect the^-vaiue 
of a radical to be very close to 2 , 0023. Nevertheless, the orbital 
angular momentum is not completely "quenched because of 
the presence of the spin-orbit coupling interactions. Normally 
there is sufficient spin-orbit coupling to leak some of the spin 
angular momentum into orbital angular momentum, and so we 



should expect the j-value to fall slightly below 2 , 0023 towards 
the value it would have if the momentum were all orbital and 
not spin (that is, towards 1 ). This is widely observed, and 
ff-values of the order 20000 and thereabouts occur frequently. 
Furthermore, the deviation from the free-spin value increases 
with the magnitude of the spin-orbit coupling constant for the 
radical, and this accords with our interpretation. For a rough 
order-of-magnitude estimate one may write the deviation from 
2-0023 as J/A, where f is the spin-orbit coupling constant and 
A a typical excitation energy of the radical {see "perturbation 
theory and "fine structure; some Rvalues are listed in 
Table 9). This mechanism can also account for the observations 
of y-values exceeding the free-spin value, which is the case 
when the electron is a member of a shell more than half full, 
but the argument is slightly more involved. Basically it is 
connected with the reversal of the order of levels when a shell 
is changed from being less than half full to more than half full. 
We know that according to the °Hund rules the levels become 
inverted (low-/ levels lie above high-/) because the sign of the 
spin-orbit coupling changes. This can be interpreted in terms 
of changing from a system in which electrons are orbiting to 
one in which holes in an otherwise completed shell are 
circulating. A hole carries, in effect, a charge opposite to an 



84 



g- value 



electron's, and the inversion of the /-levels, and the deviation 
of g to values above the free-spin value can be traced to this. 

A final kind of ff-value is the molecular g-vatus , which 
relates the rotational -angular momentum of an entire 
molecule to the magnetic moment that arises from its motion. 
Even a closed-shell molecule, such as H 2 or methane, possesses 
a magnetic moment by virtue of its rotational motion; it arises 
because of "electron slip and the consequent imbalance of the 
rotating negative and positive charges leading to a net circu- 
lating current, and thence to a magnetic moment. The 
imbalance increases as the rotational motion quickens, and so 
the magnetic moment is proportional to J, the rotational 
angular momentum. It is normally written /i — gy J, and g, 
which depends on the details of the molecular electronic 
structure, is the molecular 5-value. Mote that it is common to 
use the nuclear magneton (/i^ = eh/2m ) to define the 
molecular Rvalue: molecular magnetic moments are so small 
that this is more appropriate than using the Bohr magneton 
(using j± N makes g of the order of unity). For ammonia 
g ~ 0-53, and so the magnitude of fi when the molecule is in a 
state with J - 10 is approximately 5-3/i . 

Questions, 1 . What electronic property does the g-f actor 
determine? In an atom it is appropriate to label g with the 
value of J: why does the atomic magnetic moment depend on 
J, L, and S? Evaluate the Lande g-f actor- for atoms with S = 0, 
L=1;S = j, t = l;S=1,i = 1;S = 0, L = 2: evaluate it for 
all possible values of J in each case. The energy for a magnetic 
moment with z-component fi in a magnetic field 6 in the 
z-direction is — fj. B: evaluate the energies of the states of the 
atoms for which you have just calculated the Lande' g- factors. 
Now repeat the calculation with the false assumption that the 
^-factor for the free electron is 1 and not 2, Draw your results 
on a ladder energy-level diagram in order to see how the 
coupling of L and S modifies the magnetic properties of a 
system. Turn to "Zeeman effect for further development of 
this point. What does the^-factor measure in electron spin 
resonance? Why can g differ from 2 when the electron is a part 
of a radical? What information about the electronic structure 
of a radical cang reveal? Calculate the magnetic field for 
resonant absorption of 9 GHz radiation of (a} a free electron, 
(b) a radical with 5 = 2-0057, and (c) a radical withff= T9980. 



Quote results in gauss (G). What is the role of spin-orbit 
coupling in determing the^-value of a molecule? Why can 
someff-values be more than the free-spin value? What is the 
molecular Rvalue, and how does it arise? 
2. This question invites you to deduce the form of the Lande 
ff-factor. We consider the geometry set out in Fig. Gla with a 
magnetic field along z. We seek to express the energy of the 
system as - sryy e J.B, whereas in fact we know that the 
magnetic energy is really — 7 L.B — 27 S.B (that is, the sum 
of the orbital and spin interactions with the field). We also 
know that J = L + S. Consider L. This vector precesses about 
J, and the only time-independent, and therefore non-vanishing, 
component is the one parallel to J; this has the value L.J/I J I. 
This component now behaves like the vector [(L J)/|J|] (J/UI), 
swinging round thez-axis as J itself precesses (J/|J| is a unit 
vector along J). This vector has a projection along the magnetic- 
field direction which we may write [(L.J)/lJ l] [J.B/U I] , and 
so the only time-independent component of —7 L.B is 
— 7 e [LJ/UI 2 ) J.B. This has the form —g/f J.B, which is 
what we require. A similar expression for —27 S.B can be 
written as — g .7 J.B (find it), and so now we must show that 
g, + g, = g,, the Lande factor. Do this by noting that 
-2L.J = (J-L) 2 -J 2 -L 2 = S 2 -J 2 -L 2 , and similarly for 
S.J, and then finally replace operators of the form J 2 by their 
quantum- mechanical values J(J + 1)h 2 . Find a more exact 
form of the Lande factor using g = 2'0023 instead of a "~ 2. 

e e 

Further information. An account of the deduction of the 
Landed-factor will be found in Chapter 8 of MQM. and a further 
discussion is given in §1 1 .3 of Herzberg (1940), and §IIIF 
and §VA3 of Kuhn ( 1 969). The factor plays an important role 
in the "Zeeman effect and in determining the magnetic 
properties of transition-metal ions. For references to the latter 
see "crystal-field theory, and -ligand-f ield theory. A detailed 
account will be found in §5.6 of Griffith (1964) and §11.3 
and subsequent chapters of Abragam and Bleaney (1970). For 
theff-value of electron spin resonance see MQM Chapter 1 1 for 
a detailed discussion, including something more about <7 
exceeding 2*0023, and an account of the calculation of ^-values 
for molecules. This topic is also taken up in Chapter 2 of 
Atkins and Symons (1967), in Chapter 9 of Carrington and 
McLachlan (1967), in Chapter 7 of Slichter (1963), and in 



gaussian atomic orbitals 



85 



Chapter 12 of Griffith (1964). McLauchlan describes the 
dependence of the appearance of an electron spin resonance spec- 
trum onthep-value in Magnetic resonance {OCS 1 ). 
Molecular magnetic moments are discussed in Chapter 11, 
especially §11.6, of Townes and Schawlow (1955). 

gaussian atomic orbitals. The bore about molecular- 
structure calculations is the complexity of many of the 

electron-electron interaction-energy integrals which must be 
calculated. In a -self-consistent field calculation many of the 
integrals involve atomic orbitals based on more than one centre, 
and may involve orbitals on as many as four centres. These 
multi-centre integrals are complicated to evaluate and consume 
a great deal of time on an electronic computer. One simplifi- 
cation is to express the atomic orbital In terms of gaussian 
functions, which are basically of the form exp(— ar 2 ), instead 
of as the -Slater type of atomic orbital, which are basically 
exponential functions of the form exp(-aV). The advantage of 
this procedure arises from the fact that the product of two 
gaussians based on different centres is itself a gaussian based on 
a point lying between the centres. Therefore a complicated 3- 
or 4-centre integral can be expressed as a relatively simple 2- 
centred integral, and this can be evaluated speedily. The dis- 
advantage of the method lies in the fact that an atomic orbital 
is not well represented by a simple gaussian function, and so 
each atomic orbital has to be expressed as a sum of several 
gaussians. Therefore, although each integral is simpler, very 
many more of them need to be evaluated. 

Questions. What advantages stem from employing gaussian 
atomic orbitals, and what are the disadvantages? Show that the 
product of two gaussian functions exp(— a,r\) and exp(— a 3 r|) 
may be expressed as a gaussian function centred on a point 
between the two centres, and find the appropriate position in 
terms of a t and a t . Confirm that the procedure cannot be 
applied in the case of two exponential functions. By the 
-variation principle determine the best gaussian atomic orbital 
for a hydrogen atom; repeat the calculation using a trial 
function composed of the sum of two gaussians. In each case 
compare the true 1s-orbital with the best gaussian orbital by 
comparing the energies, and by plotting the radial dependence 
of the functions. 



Further information. See §9.2(i) of Richards and Horsley 
(1970), and §1.6F of McGlynn.Vanquickenborne, Kinoshita, 
and Carroll (1972). The method was originated by Boys 
(1950). For calculations based on the method see the bibli- 
ography prepared by Richards, Hinkley, and Walker (1971 ). 

gerade and ungerade {g and u).The German words gerade 
(even) and ungerade (odd) are added as labels to states and 
-wavef unctions to denote their behaviour under inversion 
(their parity). A simple example is the classification of the 
orbitals of the hydrogen molecule: there is a point of inversion 
symmetry of the molecule at the middle of the bond. Consider 
the amplitude of the 1 so-bonding orbital at an arbitrary point 
of the molecule and project a straight line from this point 
through the inversion centre, and travel an equal distance to 
the other side (Fig. G2). The sign and magnitude of the 
amplitude of the o-bond at the new point is unchanged, and so 
the orbital is even (g) under this symmetry operation, and is 






FIG. G2. Gerade and ungerade symmetry. 



Grotrian diagram 



written a . On the other hand, the same journey in the "anti- 
bonding orbital (1sff*) begins at a point of positive amplitude, 
passes through a -node, and ends at a point of negative 
amplitude. This orbital is odd (u) and is labelled a*. The 2p0- 
orbitals behave similarly, but the TT-orbitals behave differently 
(see Fig. G2): the bonding TT-orbital is u and the antibonding 
orbital is g. 

The g, u classification also applies to states: the appropriate 
label is obtained by determining whether the state is even or 
odd overall by considering the product of the inversion 
behaviour of its components. Thus H 2 is g because (even) X 
(even) is even; the excited "configuration a a* is u because 
g X u = u. 

When there is no centre of inversion symmetry the classi- 
fication is inapplicable: g and u labels may be added to the 
states and orbitals of homonuciear diatomics and to 
centrosymmetric octahedral complexes, but not to hetero- 
nuciear diatomics or to tetrahedral complexes. 

The classification is useful in a discussion of -selection rules 
because the only "electric dipole transitions allowed are those 
involving a change of parity; thus g — ► u and u — »■ g transitions 
are allowed, but g — > g and u — * u are forbidden. 

Questions. What is the significance of the labels g and u ? Why 
are they not applicable to heteronuctear diatomics? Which of 
the following molecules could have orbitals and states dis- 
tinguished by g and u: 2 , NO, N0 2 ,C0 2 ,CH 4 ,He, NH 3 ? 
Classify the following orbitals into g or u type symmetry: 
s-, p-, d-, f-orbitals on free atoms; Iso, 2p7T*, 3der, 3d7T, 3d7T*, 
3d5 in homonuciear diatomics. Apply g and u labels to the 
ground states of He, H 2 , and 2 . Which electric dipole 
transitions are allowed under the g, u classification? 

Further information. The g, u labelling is a group-theoretical 
classification of the parity of a state; therefore see MQM 
Chapter 5, Cotton (1963), Bishop (1973), and Tinkham 
(1964). Also see books on molecular structure, such as 
Cou [son's The shape and structure of molecules (OCS 9), 
Coulson (1961), and Murrell, Kettle, and Tedder (1965). 
King (1964) and Herzberg (1950) both discuss the role of 
g, u classification in the electronic spectra of molecules. 



Grotrian diagram. In a Grotrian diagram the energy levels of 
an atom are displayed as a ladder of lines classified into con- 
venient groups. Spectral transitions are represented by lines 
connecting the "terms between which they take place, and the 
frequency (or wave number) of the transition can be added if 
desired. Sometimes the relative intensities of the transitions are 
indicated by the thickness of the connecting lines. An example 
of a very simple Grotrian diagram is shown in Fig. G3: this is 
for the "hydrogen atom. The levels are classified according to 
the principal "quantum number, and the classes correspond to 
different values of the "orbital angular momentum quantum 
number id. The different transitions giving rise to various 
"series of lines are indicated. 



14- 



IZ- 




10- 

E/eV 

6 



4- 



2- 



n = l 



Bradierr Pfund 



PtadM 



-20000 



BohMi 



Lyman 



-40000 



■60000 



80000 



100000 



FIG. G3. Grotrian diagram for atomic hydrogen. 



*- tonaotion limit -■* 




Enenjy 
cm" 1 



- 5000 



-10000 



-IS 000 



20000 



-25000 



-SJ0O0 



-35000 



-40000 



FIG. G4. Grotrian diagram for atomic sodium. 

The Grotrian diagram for sodium is shown in Fig. G4. When 
there is more than one electron confusion sometimes arises in 
the labelling of the energy levels because in some cases the 
orbital occupied by the excited electron is used as the label, 
and in others the -term symbol, the label of the state of the 
atom. Thus in some cases the lowest state of the sodium atom 
is labelled 3s, in others it is labelled 3s 2 S, in others 3 2 S, and in 
others, reasonably but dangerously, 1 2 S, denoting that the 
ground state is the first of the doublet states. It is the pen- 
ultimate of these labels that we shall employ. 



group theory 87 

The energy is generally referred to the ground state as 
zero. Fragments of the diagrams may be magnified if it is 
desired to display the "fine structure. 

Further information. See MQM Chapter 8 for further examples 
and further discussion. Grotrian diagrams are discussed in King 
(1964), Candler (1964), who gives many examples, Kuhn 
(1962), and Condon and Shortley (1963). A collection of them 
is given in the American Institute of Physics handbook, 
p. 7-12, Gray (1972). The standard collection of energy- level 
data for atoms is that of Moore (1949 et seq.). Other aspects 
of the use of the diagrams is described under "fine structure, 
-series, and the "hydrogen atom. 

group theory. In quantum theory, group theory is the 
mathematical theory of symmetry. It puts on a formal 
mathematical basis our intuitive notions about the symmetry 
of objects, and so enables unambiguous deductions to be 
drawn about the consequences of their symmetry. Taking 
full advantage of the symmetry of a system reduces the 
amount of labour involved in calculations, and it often enables 
conclusions to be drawn without the need for elaborate 
manipulations. 

Group theory is used to find the appropriate -linear 
combination of atomic orbitals for the "molecular orbitals of a 
molecule, to classify atomic and molecular states, to determine 
the "selection rules that govern the transitions between these 
states,. and to find the -normal modes of "vibration of mole- 
cules. -Angular-momentum theory can be regarded as a 
branch of group theory. The language of group theory is 
concerned with -symmetry transformations of molecules and 
solids: with their "matrix representations, with their -character 
and their class, and with the manipulation of the irreducible 
representations. 

A brief summary of the properties of -characters, which are 
the most useful aspects in most chemical applications of group 
theory, is given in Box 4 on p 33 for those who already know 
some theory. 

Further information. See MQM Chapter 5 for a discussion of 
basic group theory, and the remainder of the book for extensive 
applications. Its connexion with angular-momentum theory is 



88 group theory 

discussed at the end of Chapter 6 in MQM. Helpful elementary Hamermesh (1962), Weyl (1930), and Judd (1963). An 

introductions to group theory are the books by Jaffe and exhaustive treatment of the symmetry properties of solids has 

Orchin (1965) and Cotton (1963), and an introductory book been prepared by Bradley and Cracknell (1972), and a small 

with many of the details filled in is Bishop (1973), Other handbook containing character tables and group properties to 

helpfuf introductions include those by Tinkham (1964), act as a notebook for group theory calculations has been 

Schonland (1965), McWeeny (1963), and, for solids, Wooster prepared by Atkins, Child, and Phillips (1970). An entertain- 

(1973), For advanced treatments see Wigner (1959), ment has been written by Weyl (1952). 



H 



hamiltonian. The name of Sir William Rowan Hamilton 
(1805—65), who was Astronomer Royal of Ireland as an 
undergraduate, was a devotee of pork chops, but not the 
husband of her who entertained Horatio Lord Nelson, is 
commemorated in quantum mechanics because he set up a 
system of classical mechanics ideally suited to the structure of 
quantum mechanics, and because he almost discovered 
quantum mechanics itself, and quite possibly would have done 
if during his life experiment had required it. For the systems 
that concern us the hamiltonian is the sum of the kinetic T 
and potential V energies (but like most things, it can be 
defined more generally, more subtly, and more powerfully). 
We give it the symbol H (some prefer 30 and so write 
H = 7-+ y. In quantum mechanics it is necessary to interpret 
observables as "operators, and interpreting the observables 7" 
and V as operators we see that the hamiltonian is the operator 
for the total energy of the system. The total hamiltonian for a 
system consists of a sum of terms corresponding to different 
contributions to the total energy. A selection of such terms is 
listed in Box 7 (overleaf). 

The -Schrodinger equation is often written in the form 
H\(j= E4>, and so it can be interpreted as an -eigenvalue 
equation, the energy E of the system being the eigenvalue of 
the hamiltonian operator, and the -wavef unctions the corre- 
sponding eigenstates. 

Questions. What is a hamiltonian? What is a hamiltonian 
operator? How may the Schrodinger equation be expressed in 
terms of the hamiltonian, and how may it be interpreted? 
Write the hamiltonian, and analyse the significance of each 



term, for the following species: a free electron; a hydrogen 
atom; a hydrogen molecule-ion; a hydrogen molecule; a helium 
atom; a carbon atom; an harmonic oscillator; an harmonically 
oscillating charge in an electric field; a mass on the end of a 
spring; and the hydrogen atom in a magnetic field. 

Further information. See MQM Chapters 3 and 4 for an 
introduction to simple ideas about hamiltonians, and Pauling 
and Wilson (1935) for another discussion. All books on 
quantum theory discuss hamiltonians, and for the classical 
background see Goldstein (1950). For further information, 
see -operators. Most hamiltonians are very complicated, and 
their eigenvalues cannot be found exactly: the three major 
approximation techniques are "perturbation theory, 
•variation theory, and -self-consistent field techniques. A 
bibliographical note about Hamilton, and his behaviour at 
breakfast, will be found in Scientific American-Whittaker 
(1954). 

harmonic oscillator. Harmonic oscillations occur in classical 
mechanics when the restoring force on a body is proportional 
to its displacement from equilibrium. A force —kx, where k is 
the *force-constant, implies the existence of a potential j** 
(because F = -3 W3x). An harmonic oscillator in quantum 
mechanics is a system with such a parabolic potential, and its 
properties can be determined by solving the "Schrodinger 
equation with a potential \kx 2 . The conclusions this leads to 
are as follows. 

1 . The energy is quantized and limited to the values 
(v + jlhojc, where «o = ik/m)^ and v is confined to the 



89 



90 



harmonic oscillator 



In general 
Free particle 



BOX 7; Hamiltonians 



tf= 7"+ V 



#/ 



0<x<£, 



H = -{tf/2m)V 2 
"Particle in a box (one-dimensional square well) 

W = -(hV2«)l 
"Hydrogen atom 

W = -{hV2m)V 2 -e 2 /47re r. 
Helium atom 

H = - {h 2 /2m\^ - (h 2 /2m}Vl - 2e 2 /47i€ r 1 - 
- 2e 2 /4we r 2 + e 2 /47reor I2 . 
Hydrogen molecule-ion (H^); fixed nuclei 

H = - ih 2 /2mW 2 , -e 2 /4m=,>r 1A -e 2 /4ire r tB + 
+ e 2 /47re r AB . 
"Harmonic oscillator 



H = ~ihV2m)(£^j +l*x 2 



Rigid "rotor 

H = - [h 2 l2l xx )f; - [h 2 i2t yy )J 2 y - ih 2 l2tjj]. 

"Magnetic dipole moment jU in field B 

H ~ — M-B; for example, ju = gy s or 7 I. 

Electric dipole moment ju in field E 
H = ~ii£. 



values 0, 1 , 2, . . .. This implies the existence of a zero-point 
energy of jhWo when the oscillator is in its lowest energy state 
with V= 0: all the energy cannot be removed from an 
oscillator. The zero-point energy may be viewed in the light of 
the -uncertainty principle: eliminating all energy, and therefore 
momentum, would imply an infinite uncertainty in position; 
but the particle is confined by the potential. The other 
implication is that the spacing between energy levels is ha^, 
and the energy of osciliation can be increased only by absorb- 



ing integral multiples of the vibrational "quantum iiGJo. The 
consequences of this are discussed further in "heat capacity 
and -quantum. 

2. The wavefunctions are simple polynomials in the dis- 
placement (the Hermite polynomials) multiplied by a gaussian 
function. The explicit form of some of the functions is given 
in Table 1 1 together with some of their more important 
technical properties. The shapes of a few of them are drawn in 
Fig. H1. In the lowest state the polynomial is simply the 
trivial factor 1, and so the wavefunction is the bell-shaped 
gaussian curve which has its maximum at the equilibrium 
position. As the "wavefunction gives the distribution of the 
particle it follows that, in the ground state, the particle 
clusters close to its equilibrium position but possesses both 
"kinetic and potential energy by virtue of the wavef unction's 




FIG. HI, The wavefunctions and energy levels of a harmonic oscillator. 
The classically accessible domains are shown bv the strong lines confined 
by the potential. View this in relation tn the upper half of Fig, F4. 



harmonic oscillator 



91 



shape and the presence of the potential. The next poly- 
nomial is essentially the factor*, and so the product of this 
and the gaussian is a wave with a -node in the centre (at x — 0). 
This corresponds to a higher energy because the wave is more 
sharply curved (higher "kinetic energy) and penetrates more 
deeply into the potential (in classical terms it swings both 
faster and further). The next function again peaks in the 
middle, but has significant accumulations of probability in 
the regions of higher potential. As the excitation increases the 
principal peaks of the probability distribution appear more 
dominantly at the limits of the distribution, at what in the 
classical treatment are the turning points of the oscillation. 
This is in accord with the classical distribution, for at the 
turning points the kinetic energy, and therefore the velocity, 
is least, and the probability of finding the particle there the 
greatest. 

3, A "wave packet may be formed when the energy state of 
the oscillator is imprecise. A wave packet with a gaussian shape 
moves from one side of the well to the other with a frequency 
COo in a manner that resembles the classical motion (Fig. H2); 




classical motion ^__^- 
motion of wave packer 




FIG. H2. Classical limit of the harmonic oscillator. 



therefore we see that the quantum spacing cjq becomes the 
frequency of the classical oscillator. This is a very good 
example of the way that quantum-mechanical principles 
underlie classical mechanics (see "correspondence principle). 
An oscillator behaves more closely in accord with classical 
mechanics as a "superposition of states becomes more 
justifiable; this increases as the quantum separation decreases. 
Therefore it can be appreciated that oscillators with fre- 
quencies of the order of 1 Hz (or 2w rad s" ! ), such as a 
classical clock, behave essentially classically, whereas periodic 
processes in atoms and molecules, with frequencies in the 
range 10 12 -10 15 Hz, behave quantum-mechanical ly. 

The importance of the harmonic oscillator is based on a 
number of features. First, oscillations in Nature are often 
harmonic to a very good approximation, and therefore the 
theory of the harmonic oscillator can be used in the descrip- 
tion of the vibration of molecules and of atoms in solids. 
This leads on to its application in the theory of "heat 
capacities and to many other properties of solids. Next, just 
as in classical mechanics, the algebra of harmonic oscillations 
is closely related to the algebra of "rotational motion, and 
therefore it is not surprising to see harmonic-oscillator algebra 
appearing in some discussions of -angular momentum. Finally, 
the harmonic oscillator is a remarkably simple creature to deal 
with because the expression for its energy is symmetrical in 
the space and momentum coordinates (both are quadratic 
functions). This feature makes it very simple to handle and is 
responsible, for example, for the equal spacing of the energy 
levels of the oscillator. The harmonic-oscillator algebra is the 
basis of the technique of "second quantization. 

Questions. 1. Under what circumstances are oscillations 
harmonic? What potential does this imply? What values of the 
energy are permitted to the quantized oscillator? What is its 
minimum permitted value? Why is it plausible that a zero- 
point energy exists for an harmonic oscillator? What is the 
separation between any pair of adjacent levels? What is the 
mathematical form of the wavefunctions? Discuss the form of 
the distribution of displacements when the oscillator is in its 
ground state, and compare it with the ground state of a 
classical oscillator and with the state of a classical oscillator 
containing the same amount of energy. How does the dis- 



92 



heat capacity 



tribution of displacements change as the oscillator is excited? 
What is the motion of a suitably-formed wave packet? Under 
what circumstances may such a packet be formed, and how 
does it relate to the motion of a classical oscillator with the 
same force-constant and mass? Calculate the energy separation 
for an oscillator of frequency 1 Hz, 10 l4 Hz (remember to 
convert these frequencies to rad s"' by multiplying by 2?r). 
Calculate the energy difference between 1 mo I of each kind of 
oscillator in its ground and excited states, and express the 
result in J mol"" 1 and in cm -1 . 

2. Use the properties of the Hermite polynomials in Table 1 1 
to deduce the selection rules for -electric dipole transitions in 
an harmonic oscillator. See °anharmonictty for a further 
development of this question. Show that the harmonic- 
oscillator Schrodinger equation can be written in the form 



a*a& 



-(X — 1 ) ^ where a is the differential operator 



(d/dy) + y, a" is (d/dy> - y, y = (mtufe/h)* 1 *, x the dis- 
placement from equilibrium, and X is related to the energy by 
E = X(h a^/2). The lowest energy corresponds to A = 1, and 
so a a^i = 0; therefore a solution is a^j = 0. Solve this first- 
order differential equation, and show that it is indeed the 

gaussian function in Table 11. 

i 

Further information. See MQM Chapter 3 for a discussion of 
the properties of the harmonic oscillator and the solution of 
the harmonic-oscillator Schrodinger equation by the method 
of factorization (in terms of annihilation and creation 
operators), and see glll.11 A of Pauling and Wilson (1935) for 
the solution by the polynomial-expansion method. The 
oscillator solution is also described in Landau and Lifshitz 
(1 958a), Schiff (1968), and Messiah (1 961 ). The properties of 
the Hermite polynomials are listed in §22 of Abramowitz and 
Stegun (1965), where numerical values will also be found; the 
manipulation of the polynomials is described in §3. 10 of 
Margenau and Murphy (1956). The relation of harmonic- 
oscillator algebra to angular momenta is described in Lipkin 
(1965), Mattis (1965), Englefield (1972), and by Schwinger 
(1965). 

heat capacity. The theory of heat capacities of crystalline 
solids began with Dulong and Petit's 'law' that all metals had a 
heat capacity of 6 cal deg -1 mol"' (25 J K" 1 mol" 1 ). Unfor- 



tunately most metals do not have this heat capacity, and none 
do at low temperatures. Nevertheless it is helpful to understand 
the reasoning that 'justifies' Dulong and Petit assertion, 
because the quantum theory is then more easily understood. A 
block of metal contains N atoms, and each can vibrate against 
its neighbours in three perpendicular directions; therefore the 
block behaves like a collection of 3/V oscillators. The -equipar- 
titton theorem states that with each oscillation at thermal 
equilibrium can be associated an amount of energy kT\ there- 
fore the total energy of the block at the temperature T is 
3NkT, or 3RT if the block is 1 mol of metal. Thermodynamics 
tells us that the heat capacity at constant volume is {%UI'dT) v , 
where U is the internal energy of the sample. In our case 
U= 3RT and so the heat capacity is C v = 3fl, the numerical 
value of which accords with Dulong and Petit's rule. 

Quantum theory warns us that the equipartion rule applies 
to a classical system and may fail for systems that, like the 
vibrations of atoms in crystals, ought to be treated by quantum 
mechanics. The root of the discrepancy lies in the inability of 
an oscillator to accept less than its full 'quantum of energy: 
this has the effect of quenching the effectiveness of the 
oscillators that constitute the sample, and therefore of lower- 
ing its heat capacity. 

The Einstein model pretends that every oscillator in the 
block has the same fundamental frequency V , and we shall 
begin with this simplified version of the true situation. Imagine 
a source of heat of temperature Fin contact with a collection 
of oscillators all having the same frequency. If the oscillators be- 
haved classically each one would be activated when the sample 
was in thermal equilibrium with the source. Each would swing 
with its natural frequency, but with an amplitude such that its 
mean energy was kT. But as the oscillator is governed by 
quantum mechanics, if heat is transferred to the metal at a 
low temperature it can be used to activate only a very small 
number of oscillators, for no oscillator can possess an energy 
less than hV E if it is to be excited at all. A little energy must 
reside in a few oscillators, and the remainder must be 
quiescent. Therefore there is an effective reduction in the 
number of oscillators in the sample, and a consequent re- 
duction of its heat capacity. At higher temperatures the energy 
may be distributed over many more oscillators and so more 



heat capacity 



93 



BOX 8: Heat capacities of solids 

Dulong and Petit 



C y = 3Lk = 3R . 



Einstein 



C, = 3R 



(O^' l expfl E /71 1 
\T/ lf1 -exp(e E /D] 2 / 
6=hvjk. 



Debye 



V 3 r6„{T 



~U2j| 4 fl/56™)7 a T<8 D 



d =hvjk. 



J> D and B D may be related to the speed of sound v. 

*£ = 3Nv 3 /4irV, 
where NIV is the number density of atoms in the sample. 



are able to accept energy: therefore the heat capacity is greater 
than at lower temperatures. At very high temperatures, when 
the energy of the block greatly exceeds the excitation energy 
of the oscillators, all oscillators are effective, and may be 
stimulated to high quantum levels, and the sample attains its 
classical heat capacity of 3ff. Box 8 gives the Einstein 
expression for the heat capacity, and Fig, H3 shows the 
pred i c ted te m pe ratu re-de pe nde n ce . 

The Debye model is a modification of the Einstein model 
and takes into account the fact that the oscillators have a 
range of fundamental frequencies from zero up to a limit V . 
One may understand this situation in two different ways. The 
first way is to regard the solid as a continuum (as a jelly). The 
jelly can vibrate at all frequencies from zero up to a very high 
value and the number of modes of oscillation that have a given 
frequency can be calculated quite simply. But the total number 
of oscillatory modes cannot exceed the total number of 
vibrational modes of the atoms that constitute the jelly. This 
number is 3/V; therefore there must be an upper limit to the 
frequency of vibration of the jelly such that the total number 
of oscillators is equal to 2N. (If the jelly were a true con- 




30 



FIG. H3. Calculated heat -capacity 
curves. 



T/e 



M 



7-0 



— i — 
90 



94 



heat capacity 



tinuum, like the vacuum, there would be no upper limit to the 
frequencies because a continuum corresponds to a system with 
an infinite number of oscillating components: for this reason 
the vacuum can support all frequencies of light.} 






FIG, H4. Oscillations in a chain of atoms, (a) the lowest frequency, 
(b) an intermediate frequency, and (c) the highest frequency. 

The other method of seeing that an upper frequency 

limit must exist is to consider Fig. H4, which shows the chain 
of atoms in a crystal vibrating relative to each other. If we 
consider only transverse vibrations (those perpendicular to the 
line of atoms) it should be clear that the vibration with the 
highest restoring force is the one in which neighbouring atoms 
are displaced in opposite directions, and the vibration with 
least restoring force is the one where all the atoms are dis- 
placed in the same direction (but by different amounts; if 
they were all displaced by the same amount we should obtain 
a translation of the block). There will also be intermediate 
modes of displacement, and therefore we can expect a range of 
restoring forces, and so a range of fundamental frequencies; 
but there will be a maximum frequency because it is impossible 
to obtain a higher restoring force than that in the situation 
where the direction of displacement changes between neigh- 
bours. This situation is illustrated in Fig. H5. 

The Debye model takes the distribution of oscillations into 
account by assessing the number of fundamental oscillations 
of each frequency v between and V Q and calculating the 



NUO 




FIG, H5. The Einstein, Debye, and experimental distribution (for Cu) 
of the number of oscillators of a frequency v. 

total contribution to the heat capacity (see Box 8 and Fig. H3). 
Since there are oscillators at lower frequency than in the single- 
frequency Einstein model the heat capacity of the Oebye 
model exceeds that of the latter, but the difference disappears 
at high temperatures when the behaviour is virtually classical. 
The heat capacity depends on the temperature and a charac- 
teristic constant known as the Debye temperature d = hv Ik; 
since the cut-off frequency is higher in rigid materials so too is 
the Debye temperature. Some representative values of 6 are 
listed in Table 12. A high Debye temperature tends to lower 
the heat capacity at a given temperature. At low temperatures 
the Debye model predicts that the beat capacity should be 
proportional to (7V0 D ) 3 , and this is often employed in 
entropy calculations. 

Further information. Chapter 1 of MQM gives details of the 
calculation of the Einstein heat-capacity formula, and some 
information about the Debye calculation. For an account of 
the Einstein calculation, and a discussion of Debye's, see 
Gasser and Richards' Entropy and energy levels (OCS 19). A 
very good discussion of heat capacities, with full details of 
calculations, will be found in Chapter 6 of Kittel (1971) and 
Chapter 2 of Dekker (1960). For further information consult 
Chapter 16 of Davidson (1962). A compilation of numerical 



Hellmann-Feynman theorem 



95 



data, including the numerical values of the Einstein and 
Debye functions and the Debye temperatures of many 
materials, will be found in §4 of Gray (1972). For the 
Debye function also see §27 of Abramowitz and Stegun 
(1965). Some Debye temperatures are given in the other 
books mentioned. 

Hellmann-Feynman theorem. The theorem states that 
the gradient of the energy with respect to some parameter 
6E/6P is equal to the -expectation value of the gradient of 
the -hamiltonian with respect to that parameter, dH/dP. In 
order to calculate dE/dP, where P is the parameter (which 
might be a molecular bond length or angle, a nuclear charge, 
or the strength of some applied field), it is sufficient first to 
calculate dH/dP, which might turn out to be a very simple 
operator, and then to ca leu fate its expectation value. The 
simplicity of this statement conceals a severe limitation 
(there has to be some catch): in this case it is the not 
unreasonable requirement that the wavefunctions used to 
calculate the expectation value must be exact. All the work 
lies in their evaluation, and faint hearts often apply the 
Hellmann-Feynman theorem even when only scruff ily 
inaccurate functions are available. This misuse can introduce 
significant errors when the wavefunctions are only slightly 
bad; so beware. 

When is the theorem used? One application is to the 
calculation of the response of molecules to electric and 
magnetic fields: see "polarizability. Another very interesting 
application is to the study of the geometry and -force- 
constants of molecules. This application stems from the 
remarkable consequence of the theorem that the force on a 
nucleus in a molecule may be calculated as a simple problem 
in classical electrostatics if the exact (quantum-mechanical) 
charge distribution is known. This deduction from the 
theorem is often called the electrostatic Hellmann-Feynman 
theorem. If we know the electron density everywhere, and 
that is known if the "wavefunction is known, then the force 
on a nucleus can be calculated by considering the 
Coulombic force that the same classical charge distribution 
exerts on a point nucleus. The •force-constant for any 
distortion of the molecule can then be calculated by 
working out the restoring force on a nucleus when the 



geometry is distorted. Note that this derivation treats the 
equilibrium geometry as a problem for classical electrostatics; 
but do not be misled into thinking that al! the nonsense about 
•exchange energy, and so on, is an unnecessary nonsense after 
all. The complication of exchange interactions is hidden, of 
course, in the difficult task of determining the correct electron 
density for a given nuclear configuration. Nevertheless, the 
theorem does enable one to remove some of the mystery about 
the shape of molecules, for in some sense the geometry can be 
understood in terms of a balance of electrostatic forces, even 
though the distribution responsible for the balance is governed 
by quantum mechanics. 

Questions. 1. State the Hellmann-Feynman theorem. Under 
what conditions is it untrue? Propose several possibilities for 
the parameter P in the calculation of various molecular 
properties. What use may be made of the theorem in the dis- 
cussion of molecular structure? Why do problems of exchange 
energy remain even though the molecular shape may be under- 
stood in terms of the forces exerted on the nuclei? Given 
that the theorem enables the forces exerted on nuclei to be 
calculated, what conditions on the forces lead to the determin- 
ation of the equilibrium geometry of the molecule? What 
might the parameter P represent in a diatomic molecule? 
What is the influence of the "exchange energy on the determin- 
ation of molecular shape according to the theorem? 
2. Prove the Hellmann-Feynman theorem by considering the 
expression E{P) ={\p(P)\ H {P)\ip{P)), where the energy depends 
on the parameter Pand so do both the hamiltonian and the exact 
normalized wavefunctions (so that (\p{P)\\{/lP)) = 1 ). Differ- 
entiate both sides with respect to P and use the fact that 
4>(P) is the eigenfunction of H(P) with eigenvalue E[P\. 

Further information. A simple account of the theorem in its 
application to electric and magnetic problems is given in MQM 
Chapter 11. For molecular applications see a thorough review 
of the subject by Deb (1973), where many more references 
will be found. More information is given in §2.6 of Slater 
(1963). For the original exposition of the theorem see 
Hellmann (1937) and Feynman (1939). 

hermitian operators. An "operator ffj is hermitian if the 
integral /d7f*S2ff is equal to the integral /dr(iV) *g. In the 



Huckel method 



Dirac "bracket notation the requirement would be for 
(fKllg) to equal {g\£l \f>*. Hermitian operators are important 
in quantum theory because their "eigenvalues are real; there- 
fore operators corresponding to physical observables must be 
hermitian. Another consequence of hermiticity is the 
-orthogonality of "eigenf unctions corresponding to different 
"eigenvalues of hermitian operators. 

Questions. 1. What is the meaning of 'hermiticity'? What 
properties stem from the hermitian nature of an operator, and 
why are hermitian operators important in quantum mechanics? 
Is the operator 'multiply by x' hermitian? Is the operator d/dx 
hermitian? Is the operator (h/i)(d/dx) hermitian? (In investi- 
gating the hermiticity of d/dx use integration by parts and the 
property that the functions f and g disappear at sufficiently 
distant boundaries,} Which of these operators might correspond 
to what physical observables? 

2. Prove that the eigenvalues of an hermitian operator are real, 
and that eigenf unctions corresponding to different eigenvalues 
are "orthogonal. Demonstrate these results specifically in the 
case of the operator fi ? = (h/iH3/30) which occurs in the 
theory of "angular momentum. 

Further information. See MOM Chapter 4 for a further dis- 
cussion of hermiticity, its consequencies, and the proof of the 
properties mentioned. See also Dirac (1958), von Neumann 
(1955), Jordan (1969), and Jauch (1968) for detailed accounts 
of the quantum-mechanical aspects of hermiticity. 

Huckel method. By taking into account the symmetry of a 
molecule in a wise way, and by making foul assumptions that 
no referee would pass nor examiner condone, Huckel devel- 
oped a simple scheme for calculating the energy of the it- 
electrons in conjugated systems. The approximations, which 
we label 1—6, are extreme, for they neglect every complicated 
aspect of the exact problem. 

1. The er-electrons are ignored; they are present implicitly 
because they are largely responsible for determining the shape 
of the molecule, but their interaction with the ff -electrons is 
neglected. 

2. All overlap is neglected. Since the overlap integral 
between neighbouring carbon TT-orbitals is about 0'25 the 



propriety of this approximation can be appreciated. (In the 
event it turns out that overlap can be included quite simply, 
and It does not affect the answers savagely.) 

What about all the complicated integrals that occur in the 
exact problem? They are classified as too difficult; but to 
eliminate all integrals would eliminate the problem. Therefore; 

3. All the integrals involving only one atom (roughly 
corresponding to the energy of an electron occupying a 
carbon 2p-orbital) are set equal to a: this is called the 
'Coulomb integral. 

4. All the integrals involving atoms separated by more 
than one bond are ignored as far too difficult. 

5. All the integrals involving neighbouring atoms are 
set equal to the same value (3: this is called the resonance 
integral. 

6. After all these approximations it does not seem worth 
"antisymmetrizing the wavef unction; so it isn't. (But it 
should be realized that in a rather ill-defined sense the effect 
of antisymmetrtzation— the role of "exchange energy— has 
been taken into account by the parameters a and j3.) 

With these approximations in hand the "variation 
method is applied to determine the best "linear combination 
of atomic n-orbitals to describe the structure of the mole- 
cule; this leads to a "secular determinant whose roots give 
the energy of the orbitals. (Each diagonal element of the 
determinant is a— £ and every off-diagonal element is zero 
except those corresponding to neighbouring atoms, which 
are set equal to p\) When the secular determinant is solved 
and the molecular-orbital energies are known the 
coefficients for the atomic orbitals may be discovered. 
This is essentially the complete solution to the problem in 
the Huckel approximation, and it is possible to deduce a 
number of molecular properties. These include the 
derealization or "resonance energy, the energy of elec- 
tronic transitions within the Tr-system, the "charge density 
on the carbon atoms, and the TT-eiectron contribution to 
the dipole moment, the "bond order, and the "free 
valence. In calculations of derealization energy a reason- 
able value of is — 0-69 eV (—67 kJ mol" 1 ), but for spectral 
transitions a better value is —271 eV (—21 900 cm -1 ). A 
sample calculation is illustrated in Box 9. 



Huckel method 



97 



BOX 9: Huckel calculation; an example 



In general: 



1. Write MO as ^ = Lc.0.. 



2. Set up -secular determinant' \H.. — ES.. I as follows: 



M if /=/ 

S/ i~ to if /*/ 

!a if / =/ 

j3 if (¥*}, 

otherwi? 



but/, /are neighbours 
otherwise, 

3. Solve \H.. - ESJ = for energies E. 

For butadiene $ = C i0j -f c 2 4>2 + c 3 3 + c 4 4 , where 
0i, ... 04 are the 2pTT-orbitals on the carbon atoms in 
CH 2 :CH-CH:CH2, The secular determinant is 

cHF 

a-f |3 

o-£ 

0/3 o-F 

If x/J = a— f this reduces to x 4 — 3x 2 + 1=0. Therefore 
the roots are 

£ = a+1-60 and £ = af±O-60 
and these are the energies of the four molecular orbitals of 
butadiene. The secular determinant can usually be simplified 
(factorized) by using symmetry arguments. See Further 
information. 



= 



The Huckel method is moderately satisfactory because 
of its reliance on the symmetry of the molecule: the 
orbitals are essentially classified according to the it- 
symmetry, and then the number of bonding and anti- 
bonding juxtapositions of overlapping 7r-orbitats is counted 
by the coefficient of j3; in this way one obtains a rough 
guide to the ordering of the energies. 

There is obviously enormous room for improvement in the 
method, and an enormous amount of work has been done with 
that in mind, A simple improvement is the inclusion of overlap 



between neighbours: this squashes the lower, bonding orbitals 

together and separates the upper, "antibonding levels without 
significantly affecting their order. The next improvement often 
employed in simple calculations is to realize that the energy of 
an electron on a carbon atom, measured by the magnitude of 
a, depends on the charge density on that atom. The Huckel 
method ignores this dependence and gives the same value of 
a to all the atoms irrespective of the focal accumulation of 
electron density. (See "alternant hydrocarbon in this 
connexion.) The a>technique seeks to repair this deficiency 
by making a depend, through a constant of proportionality Co, 
on the charge density. This is a simple example of a "self- 
consistent calculation: a Huckel calculation is first done to 
find the charge density, then each a is modified appropriately, 
and the calculation repeated with the new set of as: the 
scheme is repeated until the charge density remains constant 
through a cycle of the calculation. 

Beyond these trivial modifications of the original theory 
one encounters the semi-empirical methods which are at the 
centre of much of present-day research. These attempt to 
relax the bold assumptions of the Huckel method and to 
approach the accuracy of an exact calculation. They proceed 
by neglecting electron-repulsion integrals in a more-or-less 
rational fashion, and expressing the magnitude of the remain- 
ing molecular integrals in terms of some empirical quantity, or 
in terms of quantities calculated for atoms, or leaving them as 
adjustable parameters. Virtually all of them deal with the 
valence electrons only, and acknowledge the existence of the 
core electrons only in terms of the final choice of the para- 
meters. Most do not make the Huckel distinction between the 
a- and 7T-electrons except in so far as their symmetry is con- 
cerned; therefore the methods can be applied to molecules 
lacking planes of symmetry. Popular methods at one time 
included the Pariser-Parr-Pople (or PPP) method, which it only 
a slight improvement on the Huckel method and ignores 
virtually all the electron repulsion integrals between atoms. 
The CNDO {complete neglect of differential overlap) method 
neglects fewer integrals and is the basis of many modern 
calculations. Other initials, such as IN DO (intermediate 
neglect of differential overlap) and MINDO (modified INDO), 
represent schemes of neglecting different integrals and 



Hund coupling cases 



choosing the value of those remaining, Details of the actual 
choices and approximations in these schemes will be found 
under Further information. 

Questions. 1. List the approximations that constitute the 
Huckel scheme for calculating 7T-electron energies. What is the 
significance of the parameters a and |S? Why should a be 
expected to depend on the charge of the atom? What mole- 
cular properties may be calculated in the Huckel scheme? 
What improvements may be made quite simply? What 
improvement is it much more difficult to introduce? What is 
the major limitation on the accuracy of the semi-empirical 
methods? Why are they inferior to the best Hartree-Fock 
"self-consistent field calculations? 

2. Using the calculational scheme set out in Box 9, set up and 
solve the Huckel equations for the molecules CH 2 =CH— CH= 
CH^, CH 2 =C=CH 2 , cyclopropene, and benzene, and calculate 
the delocalization energy and charge distribution in each. In 
the case of cy do butadiene investigate the effect of including 
overlap between nearest neighbours: let the overlap integral 
be S in each case (set S = 0-25 at the end of the calculation). 

3. The tj-technique supposes that the Coulomb integral on 
atom r is related to the charge density g on r by the formula 
a r = a + {1 —g) OJp 1 , with the coefficient oj being about 1 -4. 
Apply the technique to the ally I cation starting with the 
Huckel approximation and proceeding through three cycles of 
theo>method. What is the role of the G>technique in the case 
of "alternant hydrocarbons? 

Further information. See MQM Chapter 9. Many books deal 
with the Huckel method in detail, and work through many 
examples. See Coulson's The shape and structure of molecules 
<OCS9)and §9.6 of Coulson (1961). See also Murretl, Kettle, 
and Tedder (1965), Streitweiser (1961), Pilar (1968), Salem 
(1966), McGlynn, Vanquickenborne, Kinoshita, and Carroll 
(1972), and Dewar (1969). Accounts of the developments of 
the theory will be found in these books and also in Pople and 
Beveridge (1970), Doggett (1972), and Murrel! and Harget 
(1972). A compilation of the results of such calculations has 
been prepared by Coulson and Streitweiser (1965). 
Streitweiser (1961) works through many calculations on 
organic molecules, and shows how the results may be applied 



to the prediction of chemical properties. Roberts (19616) 
works through the bare bones of the method. Parr (1963) 
has reviewed the modern theories of molecular structure, and 
his book contains a collection of some of the important 
original papers. 

Hund Coupling cases. In a diatomic molecule there are 
several sources of angular momentum, notably the "spin of the 
electrons, their "orbital motion about the axis, and the 
•rotation of the nuclear framework. The total angular 
momentum is the vector sum of all these momenta, but there 
is a variety of ways of coupling them together. The Hund 
cases are a sensible selection of a few of the possible ways of 
performing the coupling; we shall deal with the four simplest 
and most common schemes. 

(a) In Hund'scase (a) the strong electrostatic effect of the 
nuclei is respected: the axial field allows only circulation about 
the axis to survive and be a source of -orbital angular 
momentum. Through the "spin -orbit coupling any resultant 
spin of the electrons is also coupled to the surviving com- 
ponent of the orbital momentum; that is, the spins are coupled 
to the axis by a two-step process. All the electronic angular 
momentum is along the axis: the orbital component is Afiand 
the spin component Sh giving a total momentum along the 
axis of S2h with £1 = A + 2. But the nuclear framework is also 
rotating with a momentum which can be represented by a 
vector O perpendicular to the axis (Fig. H6 a). It follows that 
the total angular momentum of the molecule is represented by 
the resultant vector J. 

(b) Hund'scase (bj deals with the situation in which the 
coupling of the spin momentum to the axis disappears. This 
may occur in the case where there is no electronic orbital 
angular momentum (so there is no guidance via the spin-orbit 
interaction for the alignment of the spin) or when the spin- 
orbit coupling is so weak that the orbital angular momentum 
does not succeed in fastening it to the axis. This situation may 
arise in the diatomic molecules formed from the first-row 
atoms, especially in their hydrides. The orbital momentum is 
still coupled electrostatically to the axis, but the spin swims 
loosely around. Therefore the electronic orbital momentum 



Hund coupling cases 



99 




FIG. H6. The Hund coupling cases (a)-(d), 

and the nuciear framework momentum couple into a 
resultant N, and only then, in order to get the total 
momentum of the system, is the spin coupled in to give a 
resultant J (see Fig. H6 b). 

(c) In Hand's case (c) we go to the other extreme and 
encounter a situation in which the spin-orbit coupling is so 
strong that the spin and orbital momenta couple strongly at 
every opportunity. This passion couples them into a resultant 
total electronic angular momentum J . This J "processes around 
the internuclear axis, on which it has a component £2h. To 
this resultant couples the rigid rotation of the nuclear frame- 
work, and the whole system yields a total momentum J 



(Fig. H6 c). The spin-orbit interaction is strong in heavy 
atoms, and it is for these that case (c) is important. 

(d) In Hund's case (d) we encounter a peculiar situation in 
which the electrons are virtually independent of the orient- 
ation of the nuclear framework. The pair of nuclei churn 
around inside the molecule but the outermost electrons do 
not respond. Such a situation arises when the electron has 
been excited to an orbital outside the valence shell (into a 
"Rydberg state), for it is then so far away that the pair of 
nuclei resemble a single point nucleus. In this case the 
electronic orbital angular momentum vector combines with 
the nuclear-framework vector to yield a resultant N; with this 
combines the spin (there is very little spin-orbit coupling in 
such diffuse states) to give the total resultant J (see Fig. 
H6d). 

It should be clear that these four cases are only a few of the 
many possible; and even more are possible if the nuclei them- 
selves possess "spin. Furthermore, the cases are extreme, ideal, 
or pure cases; in any real molecule there are various com- 
petitions between different angular momenta, and for none 
can there be complete victory. The contamination of one of 
the pure Hund's cases by another is referred to as a decoupling. 

Questions. What different sources of angular momentum exist 
in a diatomic molecule? Under what circumstances does 
Hund's case (a) dominate? What happens when there is a 
vanishing orbital angular momentum in the molecule? What is 
an appropriate coupling scheme when the spin-orbit coupling 
is small? What is appropriate when it is large? What scheme is 
appropriate when an electron occupies a very diffuse orbital? 
What is actually meant by the term 'coupling'? (Consider what 
a particular coupling scheme represents in terms of energy; 
review -precession.) The quantum numbers Aand Z represent 
well-defined situations in case (a) (they are good quantum 
numbers); is that also true in case (c)? What of the quantum 
number 12? What do these quantum numbers represent? 

Further information. See MOM Chapter 1 for another look at 
the coupling schemes. See Barrow (1962), King (1964), and 
Herzberg (1950) for an account of the way the coupling cases 
are employed. There are a few more cases of interest: find 
them all splendidly discussed in Chapter 5 of Herzberg 



100 



Hund rules 



(1950). One important application is in the discussion of the 
•selection rules that govern electronic transitions in diatomic 
molecules. 

Hund rules. The Hund rules provide a simple guide to the 
ordering of the energies of atomic states. They state the 
following: 

(1) for a given "configuration the nerm with the highest 
"multiplicity lies lowest in energy; 

(2) for a given configuration and multiplicity the term with 
the highest 'orbital angular momentum lies lowest in energy; 

(3) for a given configuration, multiplicity, and orbital 
angular momentum the "level with the lowest value of the 
total angular momentum J lies lowest in energy if the con- 
figuration represents a shell less than half-filled; and the state 
with the largest value of J lies lowest if the shell is more than 
half full. 

As an example consider the configuration 1 s 2 2s 3 2p 2 and 
the terms 3 D, ' P, and ' S to wh ich it gives rise. What is the 
order of energies? Application of the first rule yields the order 
3 D< ('P, 'S). Application of the second yields 'P< *S, and 
so we can conclude that the terms should lie in the order 
3 D < 'P < l S. The third rule enables us to order the three 
levels of the triplet term ( 3 D). Since L = 2 in 3 D, and 5=1, 
the levels correspond to J — 3, 2, 1 . Since the p-shell is less 
than half full (it can hold six electrons) the appropriate order 
is Dj < D 2 < 3 D 3( and so we have a complete ordering. 

The basis of the rules lies in electrostatic Interactions for 
the first two, and magnetic interactions for the third. 

1. The first rule has an explanation that is rather more 
subtle than is found in most books. First, we can say that if 
their spins are parallel, the electrons must be in different 
orbital s ("Pauli principle), and therefore must be further apart 
than if they were crammed into the same orbital; their 
repulsion will be diminished on this account. But if we have a 
configuration 9 x 9 y . where the electrons are in different 
orbitals, it is possible for them to have either parallel or 
antiparallel spins: which lies lower, and why? The con- 
ventional, mythical explanation of why the triplet (parallel) 
configuration lies lower is that, because electrons with 
parallel spins tend to keep apart {see "spin correlation), 



their repulsion is less than in a singlet (paired) configuration. 
This is not so. Detailed calculation on some cases has shown 
that the repulsion between the electrons is greater in triplet 
states than in singlets, and that the lowering of the energy is 
due to the modification of the electron-nucleus interaction. 
Thus, in a triplet atom the electron distribution contracts, and 
is stabilized by the improved nuclear attraction: the electron 
repulsion rises because the electrons are closer together, but 
this increase does not defeat the improvement in the nuclear 
attraction. Presumably the "spin correlation helps to stop the 
electron -electron repulsion rising faster than the nuclear 
attraction. 

2. The second rule reflects the tendency for electrons 

to stay apart if their orbital angular momentum carries them 
in the same direction. Electrons circulating in the same 
direction, and therefore leading to a large total orbital 
angular momentum, can stay apart; but electrons orbiting in 
opposition will meet frequently, and so have a large 
Coulombic repulsive interaction. 

3. The third rule is of magnetic origin because the order of 
levels is determined by the 9 spin-orbit coupling interaction. 
When the spin "magnetic moment is opposed to the orbital 
moment the magnetic energy is least; but such an arrangement 
of moments implies that the two momenta are also in op- 
position, which corresponds to a low total angular momentum 
(see "fine structure, and especially Fig. F2 on p. 74). The 
inversion of the levels when the shell is more than half full re- 
flects the change in the sign of the spin-orbit coupling constant. 

Questions. State the three Hund rules. Which depend on 
electrostatic interactions and which on magnetic? In which 
does the spin-orbit interaction play an important role? What 
is the reasoning that explains the first rule? And the second? 
And the third? Why does an inversion of the levels occur when 
the shell is more then half full? Put the following terms in 
order of increasing energy: 2 S, 2 P, 4 S, 4 D. Order their levels 
appropriately on the assumption that they arise from a more 
than half-filled shell. 

Further information. See MQM Chapter 8 for a discussion. The 
Hund rules are discussed by Kauzmann (1957), King (1964), 
and Herzberg (1944). Tables of term and level energies have 



hybridization 



101 



been prepared by Moore (1949 et seq,). The rules, of course, 
sometimes fail: this may be due to "configuration interaction, 
where the presence of another configuration depresses a term 
below the position where the rules predict it should lie. The 
view that conventional explanations of the first rule are all 
wrong (including that in MQM) is based on work of Lemberger 
and Pauncz (1970), Katriei (1972), and Colpa and I slip (1973). 
See Walker and Waber (1973) for a modification to the rules in 
the case of //-coupling. 

hybridization. If an electron occupies an "orbital that has 
mixed s-, p-, d, . . . character on an atom, it is said to occupy a 
hybrid orbital, and the process of forming that orbital is 
known as hybridization. An sp-hybrid orbital, for example, is 
one composed of equal proportions of s- and p-character, and 
the electron that occupies it may be considered to have 50 per 
cent s-character and 50 per cent p-character. 

What does a hybrid orbital look like? We know that an s- 
orbital may be regarded as a spherically symmetrical standing 
wave, and that a p-orbital is a wave with two regions, one with 
a positive amplitude and one with a negative amplitude. The 
hybrid orbital is a "superposition of these two standing waves, 
and where their amplitudes are both positive there is con- 
structive interference, and where one is positive and the other 
negative there is more or less complete destructive interference. 
The superposition is lop-sided, and as shown in Fig. H7 a. An 
sp -hybridized orbital contains 33 per cent s-character, and an 
sp 3 -orbital 25 per cent. These orbitals are also illustrated in the 
figure, and it should be clear that the amount of top-sidedness 
increases through the series. Note how the node shifts as the 
hybridization changes (it passes through the nucleus in the case 
of an unhybridized p-orbital, but not for the hybrids because 
of their s-component). 

What is the point of hybridization? It will occur in mole- 
cules if there results a reduction in the energy of the molecule. 
Is the energy of a molecule reduced when its atoms hybridize? 
On first glance it might be thought unlikely. Consider, for 
example, the carbon atom, in its ground state its "configuration 
is s 2 p 2 (we ignore the deep 1s-electrons), and the 2s-electrons 
lie some way in energy below the 2p. It is natural to think that 
carbon will form bonds with its 2p-orbitals and be divalent. If, 




FIG. H7. sp -hybrid orbitals. Computed con lours for hydrogenic 2s-, 
2p-orbita!s. Note that the nodal surface is shifted from the nucleus. 



102 



hybridization 



however, we can find enough energy to promote one of the 
s-eiectrons into the p-shell the atom attains the configuration 
sp 3 : if each of the four s- and p-orbitals hybridize to form four 
sp 3 -hybrids we can envisage a situation in which each electron 
occupies one of the hybrids. The form of these is dear from 
Fig. H7: they are strongly directional, and a little calculation 
shows that they are directed towards the four apices of a 
tetrahedron (Fig. H8) (see "equivalent orbitals). Nevertheless 
we still have not recovered our initial energy investment which 
was used to promote one of the s-electrons. At this point we 
can appreciate that where we shall recover the energy is in the 
four bonds that we are able to form. Because of the strong 
directional properties of the hybrids they will have a very 
efficient overlap with the four neighbouring atoms, and we 
can expect four strong bonds. This is where we draw our 
reward: more than the initial promotional energy is regained 
by the formation of four strong (and 'equivalent) bonds. 




FIG, H8. Tetrahedral sp -hybridization (for example, in CH4I. 

The case of ammonia (NH 3 ) helps us to appreciate another 
source of improvement in energy (Fig. H9). The nitrogen 
atom has the configuration s 2 p 3 , and so we can expect to 
form a pyramidal ammonia molecule by attaching one 
hydrogen atom to each half-filled nitrogen 2p-orbital. Suppose 
an s-electron were promoted into the p-shell; would the energy 




FIG. H9, Ammonia and hybridization; the lone pair is drawn in colour. 
In (a) there is no promotional energy, but high repulsive interactions 
and moderate overlap; in (c) low repulsive interactions, strong overlap, 
but high promotional energy; in (b) is illustrated the compromise. Only 
two bonds are illustrated. 



of the molecule be lowered? First we invest promotion energy 
by agreeing to treat all the s- and p-orbitals on the same 
footing. We form sp 3 -hybrid orbitals with a tetrahedral dis- 
position and strong directional properties. Then we feed in five 
electrons and add on three hydrogen atoms. The resulting struc- 
ture consists of three 0-bonds, each formed from a strongly over- 
lapping sp 9 -hybrid and a hydrogen Is-orbital (and each con- 
taining two electrons), and a further two electrons to form a 
"lone pair. With three of the bonds we get a return on the 



hydrogen atom 



103 



energy invested because of the excellent overlap; but we also 
recover some energy from the lone pair, because it is con- 
centrated in a region where its electrostatic repulsive inter- 
action with the three rj-bonds Is minimal; furthermore, the 
larger bond angles yield a reduced H-H repulsion. The actual 
structure of ammonia is a compromise between the amount of 
energy required to promote an s-electron into a p-orbital, and 
the energy that can be obtained by improving overlap and 
diminishing nonbonding interactions. In the event it turns out 
that the molecule is some way between the two extreme 
structures. {This can be determined by measuring the bond 
angles: in the pure p-orbital case the H-N-H angle is 90°, and 
in the pure sp 3 -orbital case it is 109° 28'; the experimental 
value is 103°.) 

Carbon is well suited for hybridization because its pro- 
motional energy is quite small (the- promoted s-electron can 
enter an empty p-orbital); therefore it shows a range of 
hybrids, depending on the demands made under the reaction 
conditions. In CH 4 and its homologues the hybridization is 
almost pure sp 3 ; in alkenes and in aromatic molecules there is 
a planar, triangular array of sp 2 -hybrids and the remaining 
2p-orbital of this promoted state forms the jr-bond. In 
acetylenic compounds (alkynes) the fj-bond is sp-hybridized, 
the remaining two 2p-orbitals forming the 7T-bonds. 

A list of the symmetries of hybrid orbitals is given in 
Table 13. A discussion of the formation of hybrid orbitals will 
be found under "valence state. 

Questions. 1. What is meant by the term 'hybridization? 
Sketch some hybrid orbitals {especially sp, sp 2 , sp 3 , sd, sd 2 ). 
Why can hybridization lower the energy of a molecule? 
Describe the formation of CH 4 , NH 3 ,and H 2 0, and explore 
the contributions to the energy which are modified by 
hybridization of the central atom. Why is hybridization of 
special importance in carbon? 
2. The four sp 3 -hybrid orbitals may be expressed as 



s + p x +p k + p,;s 



P z ;s-p x 



p "+" p ; 



s p x + fV ~~ p ?" Snow tnat tnese are directed towards the 
apices of a regular tetrahedron, and that they are mutually 
-orthogonal. Express a hybrid orbital in the form 
as + (3{&p x + mp + np z ), where £, m,n are the direction 
cosines of the bond that we hope to form, and a 2 + ft 1 = 1 . 



Take another equivalent orbital pointing along the direction 
$.', m', ri and otherwise of the same composition. From. the 
condition that the functions are -orthogonal and -normalized 
deduce that a. and j3 are related to the angle between the 
bonds by a 2 i$ 2 = - cos 0, and 1/0 2 = 2 sin 2 ^3. Discuss the 
form of the hybrids when 8 is 180°, 90°, and the angle of a 
tetrahedron. Plot s-character as a function of 8. Show that the 
effect of hybridization is important only when the orbitals 
involved are of approximately the same size. Proceed by 
writing a and in the preceeding expression as sin£ and cos£ 
respectively (so that a 2 + fi 2 = 1 is satisfied automatically), 
and write Hp x + mp + np. as the orbital p. Find an expres- 
sion for the overlap with a neighbouring orbital tpin terms 
of the overlap integrals S ■ <s|$ and S — <p|^>), and find the 
expression for the maximum value of S as a function of £. The 
answerisS = [S 2 + S I ) w . This shows that S differs 

max s p 

appreciably from S ot S only if S ~ S ; in the event of 

. » P ' s,, P 

these overlaps being equal 5 = 2 A S . 

Further information. See MQM Chapter 9, and especially 
Appendix 9,2 where a fuller discussion of hybridization, 
equivalent orbitals, and localized orbitals is given, together 
with the solution to Question 2. The importance of 
hybridization in molecular structure is described by 
Coulson in The shape and structure of molecules (OCS 9) and 
in Chapter 8 of Coulson (1961). See also Murrell, Kettle, and 
Tedder (1965), Pilar (1968), Kauzmann (1957), and McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972), The last 
book describes the formation of hybrids in some detail and 
evaluates a number of useful overlap integrals. The group- 
theoretical formation of hybrid orbitals is described in MQM 
Chapter 9. 

hydrogen atom. The hydrogen atom, consisting of an 
electron surrounding a proton, was one of the fences that 
classical mechanics failed to take, and one of the remarkable 
successes of quantum mechanics and its later developments. 
The spectral observations on the hydrogen atom showed 
that it emitted and absorbed light at a series of well-defined 
frequencies (Fig. H10). In 1885 Baimer spotted a relation 
satisfied by the frequencies that lie in the visible region of the 
spectrum. This Baimer series (Fig. H10) fitted the formula 



104 hydrogen atom 

1 05 000— 



too 000- 



95 000 J 

90 00C- 
85 000 



25 0CO 



20 000- 



I50O0- 



10 000— 



5 000- 



■ 97-253 - 



(02-572 



I2I567 -lL, 



Lyman 



-43*05 - 
-486-0 - 



-feSfc.28 -K 



H. 



Balmer 
(visiMe region) 



-820 4 -, 



-093-8 — 

-I28I-8 - 
-I458-4 — 



Ftosehen 



I875J — ' 



—2278-8 - 

^262 r ' 



Bracket? 



-405I-2 
—74000 . 



ipfjnd 



FIG. H10. The spectrum of atomic hydrogen. 



V= ff(1/2 2 - Mn\), with n i an integer greater than 2. It is 
tempting to speculate on the existence of other series of lines 
in which 2 2 is replaced by n\,n 2 another integer, and in due 
course Lyman discovered his series {in 1914 and the ultra- 
violet) correspondingtor^ = ],Paschen his [n 2 = 3; in 1908 
and the infrared), Brackett his (n 2 = 4, in 1922 and the 
infrared), and Pfund his (n 2 - 5, in 1924 and the far infrared). 
Those, for the moment, are the facts. 

°Bohr constructed a theory of the hydrogen-atom structure 
which drew on Rutherford's nuclear model and the quantum 
hypothesis; but the model had defects, and it was replaced by 
a deduction of the structure from the "Schrodinger equation 
by Schrddinger himself in 1926. A principal feature of this 
theory is that the energy of the atom is "quantized and limited 
to values given by —R in 2 , where n is an integer greater than 
zero (the principal "quantum number) and R the "Rydberg 
constant, which is a collection of fundamental constants 
having the value 109 677 cm" 1 . A transition from the level rti 
to the level n 2 involves an energy change in splendid accord 
with observation (see the -Grotrian diagram. Fig. G3 on p. 86). 
The features of the structure are as follows: 
1. The electron is distributed around the nucleus in 
"orbital s. Each atomic orbital can be distinguished by a set of 
three "quantum numbers. These are the principal quantum 
number n, the 'angular-momentum quantum number 
(occasionally called the azimuthai quantum number) 9., and the 
magnetic quantum number m^. The principal quantum number 
may take any integral value greater than zero, the azimuthai 
quantum number may take any integral value from zero to 
n — 1, and the magnetic quantum number can take any 
integral value between £ and — £. The energy of the state is 
determined solely by the value of the principal quantum 
number and is given by — R H ln 2 . (More concerning the 
"Rydberg constant will be found under that heading.) The 
angular-momentum quantum number determines the 
magnitude of the "orbital angular momentum of the electron 
about the nucleus through the formula [£(£ +- 1)] ' A h. The 
magnetic quantum number determines the orientation of this 
angular momentum in space, and, in accord with the general 
properties of "angular momentum, the component of angular 
momentum of the electron about an arbitrarily selected axis is 
equal to mnh. 



hydrogen atom 



105 



color ifude 



oiimufh 




FIG, H1 1. The spherical coordinates used to discuss the position of the 
electron in hydrogen. 

2. The orbital corresponding to the state with quantum 
numbers n, %, and m^ is in general a function that depends on 
the distance r of the electron from the nucleus, the colatitude 
(the angle away from the atom's north pole), and the azimuth 
(the electron's longitude) {Fig. H11), and may therefore be 
written $ n faJfi 0, (j>). This orbital function may be expressed 

as a product of a function dependent solely on the radius and 
of one dependent solely on the angles: \}/{r, 6, <j>) = 
R ni^ Y 9jnJ^' $' The an 9ular functions Y^ (0$) are the 
•spherical harmonics. 

2(a) When & = the orbital is isotropic because V [6, 0) 
is a constant (1/2JT* 4 ) and independent of the angles. The 
orbital in this case is known as an s-orbital, and an electron 
that occupies it is distributed symmetrically around the 
nucleus (Fig. HI 2). Since fi = the angular momentum of an 
electron in this orbital is zero; and in this connexion it is also 
important to note that there no angular "nodes in the orbital 
(the number of such nodes determines the -angular 
momentum). 

2(b) When £ = 1 the orbital is ap-orbital, and its angular 

dependence is given by the function Y, „, (0, d>). This is not a 

imp 
constant, but has its maximum amplitude along particular 

axes in space. When mg = the function is Y m (#, <j>), which 

might look fearsome, but in fact is simply the function cosfl; 

this has its extreme along the z-axis, where = or 180°, and 

so an electron that occupies this orbital is most likely to be 

found in regions concentrated along the z-axis {Fig. H 12): 




FIG, H12. Representation (by approximate boundary surfaces) of 
s-, p-, d-, and f-orbitals. 



106 



hydrogen atom 



for this reason it is referred to as ap^-orbital. The other 
possible values that m^ may have (when 8 = 1 ) are ± 1 , and in 
both these the dependence on $ is as sinO: this implies that 
both are concentrated predominantly in thexy-plane and have 
zero amplitude anywhere on the z-axis. Both Y x t and V,_i 
are complex functions, {see Table 23 on p. 282), but the sums 
and differences are real, and so these combinations are easier 
to depict. y 1( i + Y } ., is the simple function sinfl cos0, and 
so it is a standing wave concentrated along thex-axis, because 
there both sin 6 and cos0 are maximal: this is ap -orbital. The 
other combination. K,,- Y \,-i is the function sin0sin0, 
which is a standing wave concentrated along the /-axis, and 
therefore is referred to as a p -orbital. These have the same 
shape asp^, but differ in orientation (Fig. HI 2). With each 
p-orbital there is associated an angular momentum of 
magnitude [1(1 + 1)f /j ri,or (/ 2) h, and it should be clear 
how the different values of the quantum number m« dis- 
tinguish the different orientations of the distributions of the 
electron, and therefore the different orientations of the 
"orbital angular momentum of the electron occupying that 
orbital (for example, when m^ — ± 1 the electron is largely in 
the xy-plane and so most of its angular momentum is about 
the z-axis). 

2(c) Next up the scale of S values lie the d-orbitals with 
£ = 2. Of these there are five (mn = 2, 1, 0, —1, —2), and 
although the five functions Y im (0, 0) are complex it is 
possible to select five real combinations (although it is not 
possible to choose five similar shapes). The d j orbital is 
concentrated in thez-direction, and the notation z 2 may be 
construed as implying a stronger concentration along z than 
that of the p-orbital denoted p The d -orbital has its major 
concentrations along the bisectors of the x- and z-axes, hence 
the notation, and so there are four lobes (Fig. H12}. The d - 
and the d-orbitals are similar in form to d but, as the 

X Y xz 

notation suggests, are concentrated along the bisectors of the 
y- and z-axes and the x- and y-axes, respectively. The fifth 
d-orbital is d 2 2, and this peculiar notation implies that 
the lobes resemble the lobes of the preceding three orbitals 
but are directed along the x- and /-axes. 

2(d) The seven f -orbitals, corresponding to S. = 3, and 
therefore to a high angular momentum of magnitude (/12)ft, 



are the spherical harmonics Y 3m [8, 0), and seven possible 
real combinations are illustrated in Fig. H12. It is very un- 
common to encounter orbitals of higher momentum, but their 
form may be deduced quite simply because the angular 
dependence of all the "spherical harmonics is known, 

3. The radial dependence of the orbitals is contained in the 
function R n %{r), and it reveals how closely the electron clusters 
around the nucleus. It is of major, but not of sole, importance in 
determining the potential energy of the electron and therefore 
the energy of the atom. The mathematical form of the 
functions had been encountered long before the Schrodinger 
equation was solved, for they are essentially the associated 
Laguerre functions. Do not be put off by the complexity of 
the name: they are simple polynomials in r (Table 14). 

All the radial functions tend rapidly towards zero as the 
radius becomes large, and this reflects the improbability of 
discovering the electron at great distances from the nucleus. 
All except the functions R n Ar) vanish at the nucleus, and so 
if the electron occupies an o'rbital with S. ¥^ there is zero 
probability of discovering the electron actually at the nucleus. 
The R n Q (r) functions are peculiar because all are non-zero at 
the nucleus, and so all predict a non-vanishing probability of 
finding the electron there. Since It = for these functions we 
conclude that only for s-orbitals is there a non-zero probability 
of discovering the electron at the nucleus. The physical basis 
of this difference can be understood by recalling that an 
electron in an s-orbital has zero orbital angular momentum, 
and so drifts in toward the nucleus, whereas for all other 
orbitals (£ > 0) the angular momentum provides a source of 
centrifugal force that flings the electron away from the 
nucleus so strongly at short distances that the attractive 
Coulombic potential is overcome. Between the nucleus and 
infinity the radial wavefunction oscillates a varying number of 
times: the number of "nodes in the function R n is n - £ — 1. 
The radial behaviour of the functions is depicted in Fig. H13, 
and it is helpful to remember that the 1 s-orbital {n = !,£ = 0), 
which is the lowest-energy orbital, is simply a decaying 
exponential function which falls towards zero from a finite 
and non-zero value at the nucleus. 

4. The complete orbitals are depicted in a few cases in 
Fig. H14 and listed in Table 15. The shapes of the orbitals 



hydrogen atom 



107 




FIG. HI 3. Radial vuavef unctions for some states of hydrogen. 
Probabilities Inot radial-distribution functions} are shown in colour. 



themselves and the probability distributions (which, according 
to the Born Interpretation, are the squares of the orbital 
functions) have more-or-less the same shape, but the 
directional dependence is more pronounced. It is inconvenient 
to draw such complicated diagrams to illustrate aspects of 
atomic and molecular structure and, in accord with the dis- 
cussion of -orbitals, it is common to draw a boundary surface 
within which there is a high probability of finding the electron; 
this was done to obtain Fig. H 12. Note that the shape of the 
surface depends on whether one is attempting to catch a 
particular proportion of the amplitude or of the probability. 
Beware. The connexion of the boundary surfaces drawn here 
should be compared with the depiction of the spherical 
harmonics by a similar device on p. 221. The details of the 
shape of the orbitals are generally of small importance, and 
so boundary surfaces, which are normally drawn with scant 
regard for precision, like those in Fig. Ht2, should be inter- 
preted as rough boundaries for regions of concentrated 
amplitude. 

5. The most remarkable feature about the structure of the 
hydrogen atom is the dependence of the energy on only the 
principal quantum number and its independence of the orbital 
angular momentum quantum number K. This means that every 
state of the atom with principal quantum number n has the 
same energy irrespective of the values of C and m^. Since for a 
given n the value of £ may range from to n — 1 , and for each 
£ the value of m^ may range from — 1 to C, it follows that a 
state with quantum number n is/7 2 -fold "degenerate. This 
peculiar degeneracy, called by the shallow accidental, is a 
consequence of the very high symmetry of the central 
Coulomb potential, and is lost when the purity of the 
potential is destroyed by the presence of the other electrons in 
a many-electron atom (see "penetration and shielding), 

6. The energy of the ground state is worth discussing in 
greater detail. On classical grounds one expects the electron 
to spend as much time as possible in the vicinity of the 
nucleus, because by so doing its potential energy is minimized. 
The ideal position of the electron is that of contact with the 
nucleus. But the ground state of the hydrogen atom, an 
electron in a 1s-orbital, has an electron distributed in regions 
close to the nucleus, but certainly not wholly confined to it. 



108 



hydrogen atom 




hydrogen atom 



109 



FIG, H14. Amplitude contours of some hydrogen atomic 
orbitals. 




What repels the electron? The classical answer is its angular 
momentum and the concomitant centrifugal force, and this 
was the basis of -Bohr's model. But with the s-orbital there is 
associated no angular momentum, and so this cannot be the 
explanation. The answer is found in the implication of the 
shape of the radial wavef unction and the connexion between 
its curvature and the 'kinetic energy. 

In Fig, H15 are shown three possible distributions of the 
electron in a 1 s-orbital. In Fig. H 15c the electron is strongly 
confined to the vicinity of the nucleus, but the expense of 
doing so is discovered in the sharpness of the curvature of the 
wavefu notion: this corresponds to a very high component of 
kinetic energy associated with the radial direction. Fig. H15a 



shows a situation in which the kinetic energy has been lowered 
by permitting the electron a more diffuse domain; but here 
the expense arises in the high potential energy of a situation 
in which it is allowed to be at a considerable distance from the 
nucleus. It is clear that there exists a compromise distribution, 
as shown in Fig. H15b, in which the electron achieves a 
balance between a moderate kinetic energy and a moderate 
potential energy: this is the ground state of the hydrogen 
atom. This situation may also be interpreted in terms of the 
-uncertainty principle (see Questions). 

The major features of the structure and spectrum of the 
hydrogen atom are well explained by quantum mechanics; but 
complications are normally discovered when stones are over- 



110 



hydrogen atom 




FIG. H15. Curvature, potential energy, and compromise determining 
the ground state of atomic hydrogen, 

turned and the ground inspected in more detail. A closer 
scrutiny of the spectrum reveals that the lines depicted in 
Fig. H10 do in fact have a very fine structure, appropriately 
known as »fine structure. This can be explained, as described in 
that section, in terms of the coupling of the spin and orbital 
momenta, and the energy of the interaction, and the splitting, 
emerged naturally as a consequence of the "Dirac equation. 
Nevertheless, a slight discrepancy remained even after the 
application of that beautiful theory, and the Lamb shift, a 
splitting between the levels 2 S % and 2 P^ (which on the Dirac 
theory are strictly degenerate), was accounted for only when 
the hydrogen atom was treated in terms of -quantum electro- 
dynamics. Nevertheless, the closeness of the predictions of 
quantum theory and the experimental results was a triumph of 
the theory, and in its turn the explanation of the minute Lamb 
shift (a discrepancy of only 0-033 cm" 1 } is a triumph of the 
more recent modifications of quantum theory. 

Questions. 1, What is the evidence concerning the structure of 
the hydrogen atom? Estimate the highest frequency transition 
in each of the five spectral series. Street lamps containing 
sodium often glow red before they turn yellow on account of 
the hydrogen they contain: what transition is responsible? 



Calculate the "ionization potential of the hydrogen atom from 
the frequency of the lines in the Lyman series (Fig. H10). What 
is the significance of the quantum numbers n, %, and mn used 
to denote an orbital in the hydrogen atom? What is the general 
form of the orbitals? What are the features of the s-orbitals? 
What distinguishes them from all the other orbitals? How 
many p-, d-, f-, and g-orbitals are there in the case of n = 1, 2, 
3, 4? How many nodes does each orbital have? How many 
radial nodes, angular nodes, and total nodes does an orbital n, 
£, mg possess? What is the connexion between the number of 
nodes and the orbital angular momentum? What is the signifi- 
cance of the number of radial nodes? What is represented by 
the curves that are often drawn to denote atomic orbitals? 
How many different orbitals correspond to the energy 
— /? H /n 2 ? Does the value of mg affect the energy? What is the 
physical interpretation of the structure of the hydrogen atom 
in its ground state? How would the argument be modified to 
account for the structure of the similar species He + ? Discuss 
the ground state in terms of the uncertainty principle: con- 
sider the effect on the "momentum, and therefore on the 
"kinetic energy, as the electron is confined more severely to 
the vicinity of the nucleus. Discuss the effect on the atomic 
orbitals of modifying the nuclear charge: what happens to the 
orbitals as the atomic number of the nucleus increases from 
1 to 4? What deficiencies are there in the quantum-mechanical 
description of the hydrogen atom? 

2. From the Tables (p. 275) plot the radial wavefunction for 
the Is-, 2s-, 2p- r and 3d-orbitals of the atom. Demonstrate 
explicitly that these functions are "normalized. Plot the 
probability distribution as a function of r and note that the 
most probable position for finding the electron in an s-orbital 
is at the nucleus. Plot the "radial distribution function for the 
Is- and 2s- orbitals. Calculate the most probable radius for 
finding the electron in the Is-, 2s-, and 2p-orbitals. Interpret 
the different values physically. Calculate the most probable 
radius of the atom as a function of Z. At what radius does the 
1s-etectron lie, most probably, in the atoms He, C, F, and U? 

Further information. See MQM Chapter 3. For the explicit 
solution of the hydrogen atom see Pauling and Wilson 
(1935) for the series solution, and Kauzmann (1957), Davydov 
(1965), Messiah (1961), and Schiff {1968), For an account by 



hyperconjugation 



111 



the much more elegant method of factorization of the 
Schrddinger equation, see Infeld and Hull (1951), Green 
(1965), and Englefield (1972). For a discussion of why the 
degeneracies are not accidental, see MQM Chapter 3, 
Englefield (1972), Bander and Itzykson (1966), Mcintosh 
(1959, 1971), and Fock (1935). For further depictions of the 
orbitals, and their relation to the classical orbits, see White 
(1935). See also Herzberg (1944). For a discussion of the 
spectrum of atomic hydrogen, see MQM Chapter 3, King 
(1964), Kuhn (1962), Herzberg (1944), and Series (1957). 
Through its bibliography the article by Mcintosh (1971) will 
set you on a path through abundant fascination: through 
accidental -degeneracy, projections of hydrogen atoms on to 
hyperspheres and hyperhyperbolas, and the four- and two- 
dimensional atom. Theoretical and experimental data for the 
hydrogen atom are collected in Table 16. 

hyperconjugation. The -overfap of o-orbitals and jr-orbitals 
such as the overlap of the methyl C— H o-bonds with the 
aromatic jt -orbitals in toluene, is called hyperconjugation, or 
the Baker-Nathan effect. It constitutes a mechanism whereby 
the methyl group can behave as an electron donor, and its 
consequences include increased electron density at ortho and 
para sites, and concomitant effects on aromatic reactivity. 

The quantum-mechanical description considers the three 
fj-bonds of the methyl group, or, what is equivalent for the 
present purpose, the three hydrogen 1s-orbitals, as a unit 
from which three group orbitals may be constructed 
(Fig. H16). If the in-phase combination of the three orbitals 
is taken, an orbital embracing all three may be formed; this 
orbital has cylindrical symmetry about the C— C bond 
direction, and so has no net "overlap with the TT-system. Two 
other combinations of the three atomic orbitals may be made, 
and both have a single mode. One has its node perpendicular 
to the ring, but the other's lies in the plane of the ring, and 
therefore has the same symmetry with respect to the C~C 
bond as the JT-orbital on the ring carbon atom. It can therefore 
overlap it, form a very weak bond, and provide a means of 
mixing the C— H bond electrons with the TT-electrons. 

Further information. See §2.3 of Coulson (1961) and §5.7 of 
Streitweiser (1961) for many references. See also Murrell, 



plane of ring 




FIG. HI 6. Group orbitals on CH3. Only (c) has non-zero net overlap 
with the Jr-orbital on the neighbouring aromatic ring, 

Kettle, and Tedder (1965) and Salem (1966). A discussion 
of the evidence and consequences for hyperconjugation are 
described by Baker (1952) and in a conference proceedings on 
the subject. Baker (1958). Group orbitals are discussed by 
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972), 
especially §2.8. Good evidence for hyperconjugation comes 
from "electron spin resonance: see Symons (1963), Ayscough 
(1967), and Bolton, Carrington, and McLachlan (1962). 

hyperf ine interactions. A hyperfine interaction is an inter- 
action between an electron and a nucleus other than their 
point-charge Coulombic interaction. One may distinguish 
between electric and magnetic hyperfine interactions: the 
former arises because the nucleus may have an electric 
°quadrupole moment, and the latter because it may have a 
"magnetic dipole moment. We consider them separately. 

Magnetic hyperfine interactions. A nucleus with no n -zero 
"spin possesses a magnetic dipole moment; this dipole gives 
rise to a magnetic field in its vicinity, and with this field the 
magnetic moments of the electrons of the molecule may 
interact. Consider the important case where the molecule 



112 



hyperfine interactions 



contains a single unpaired electron. This electron possesses a 
■magnetic moment by virtue of its spin, and this moment 
interacts with the dipolar field of the nucleus by a conven 
tional dipole-dipole interaction. The interaction energy 
depends on the relative orientations of the spin magnetic 
moments (just as in the case of two small bar magnets) and 
also on the relative disposition of the two spins. 




FIG. HI 7. Dipole-dipole interaction of parallel moments. 

Suppose that there is present an externally applied 
magnetic field (as in the -electron spin resonance experiment) 
which is strong enough to hold parallel the two magnetic 
moments of interest. The magnetic interaction energy now has 
the classical form for the interaction of two parallel dipoles, 
and this depends on/- asr~ 3 and on the angle & as 1— 3cos 2 
{Fig. HI 7 and Box Son p. 50}. If the unpaired electron 
occupies an s-orbital on the magnetic nucleus it is spread 
isotropically about it {Fig. H18 a): the net field it experiences 
is zero, and so there is no dipole-dipole interaction in this 
case. If the electron occupies a p-orbital on the magnetic 
nucleus its distribution would not be isotropic, and in fact it 
would sample some regions of the nuclear dipole field more 
strongly than others (Fig. H18 b). In such a case the energy of 
interaction does not vanish. 

For a given electron spin orientation (expressed by the value 
of the -quantum number m ) the energy of interaction depends 
on the nuclear spin orientation m r Since a nucleus of spin / 




FIG. H18, Field from a nuclear magnetic moment: (a) an s-electron 
samples positively and negatively directed field equally; (b) a p-electron 
samples the positive field more than the negative (in this orientation). 

may have 2/ + 1 orientations the electron may experience one 
of 2/ + 1 different values of the local magnetic field. In an 
"electron spin resonance experiment this leads to a splitting of 
the spectrum into 2/ + 1 lines with a separation determined by 
the strength of interaction between the two magnetic dipoles. 
If the molecule (radical) is rotated relative to the applied 
magnetic field, and the moments retain their original 
projections m and m., the electron samples different regions 
of the nuclear hyperfine field because the p-orbital distribution 
is carried around by the rotating nuclear framework. It follows 
that the dipole-dipole hyperfine interaction is anisotropic 
{depends on orientation of the molecule). If the rotation is 
very fast, as it is for a molecule in a liquid, the rotation causes 
the electron spin to sample all the different nuclear hyperfine 
fields: this spherical average is zero and so the hyperfine struc- 
ture disappears from the spectrum. 

Hyperfine fields at the position of an electron may amount 
to tens of gauss. Typical values are given in Table 9. These 



hyperfine interactions 



113 



values are for a p-electron confined completely to the atom 
containing the magnetic nucleus; therefore if a smaller value 
is measured in an experiment the ratio gives the contribution 
of that orbital to a molecular orbital extending over the mole- 
cule. 

Although the above analysis predicts that the magnetic 
hyperfine structure of electron spin resonance spectra should 
disappear in fluid solution, experimental observation shows 
that this is far from the truth, for there usually remains a 
splitting of lines which is in many cases considerably larger 
than the anisotropic hyperfine structure just described (see, 
for instance, Fig. E4 on p, 63). The interaction responsible 
for this is the "Fermi contact interaction. It was an approxi- 
mation in the preceeding discussion that the nucleus gave a 
truly dipolar field. That would be sufficiently accurate if the 
nucleus were vanishingly small or if an electron could never 
approach it sufficiently closely to realize that it was not an 
infinitesimal point. The nucleus does have an extension (its 
diameter is of the order of 10" 14 m) and if the electron is in an 
s-orbital it may approach the nucleus very closely. When it 
gets there it discovers that the field is not a pure dipolar field 
(see Fig, F1 on p. 72). Let us take the view that the electron 
actually penetrates the nucleus, and let us replace the nuclear 
magnetic moment by an equivalent current loop with the same 
radius as the nucleus. Outside the loop the field does 
average to zero when the electron spin samples it spherically; 
but inside the field is all in the same direction, no cancellation 
occurs, and the spin magnetic moment interacts with the non- 
vanishing average of the field. This interaction is isotropic 
because it is independent of the orientation of the remainder 
of the nuclear framework. It is unique to s-electrons because 
all others have a -node running through the nucleus and so 
cannot penetrate it. 

The Fermi contact interaction may be very large: some 
typical magnetic fields experienced by the electron by this 
mechanism are also listed in Table 9. Since it is characteristic 
of s-orbital character the magnitude of the observed isotropic 
splitting may be used to determine the amount of s-electron 
character in a molecular orbital. 

Now let us turn off the external magnetic field, and let all 
the competition for magnetic coupling be between interactions 
within the molecule. The nuclear magnetic moment may 



couple to the magnetic fields from various sources within the 
molecule, and the energy of this interaction will appear in the 
spectrum (particularly the rotational spectrum) as a splitting 
of lines; molecules with different nuclear spin states will give 
rise to slightly shifted spectra, and therefore the observed 
spectrum will be the collection of lines from all the different 
molecules in the sample. The other sources of magnetic 
moment in the molecule include the electron spin (which now 
couples differently because there is no applied field holding it 
parallel to some laboratory axis), the electron orbital 
magnetic moment, and the 'molecular magnetic moment. 

Electric hyperfine interactions. It is not possible for a 
nucleus to have an electric dipole moment (on grounds of 
symmetry), but it may have an electric 'quadrupole moment 
if its spin quantum number / is 1 or larger. The presence of an 
electric quadrupole moment implies an asymmetry in the dis- 
tribution of charge in the molecule, and there are two possi- 
bilities. These are illustrated in Fig. H19. In one, the prolate 
case, there is an excess of positive charge in the polar regions 
of the nucleus (with respect to the axis of spin), and a com- 
pensating slight relatively negative band in the equatorial 
zone. In the oblate case the distributions are reversed. An 
electric "quadrupole moment interacts not with an electric 
field itself but with the field gradient. If, therefore, there is an 
electric field gradient at the quadrupolar nucleus, the energy 
of the molecule will depend on the orientation of the nucleur 
in the molecular framework, and so we arrive at another 
hyperfine interaction. 








distribution 
of pcwitive choge 



© 




low eneigjr high enenjy 

FIG. H 19. A nuclear eluctric quad ru pole in a field gradient. 



114 



hyperfine interactions 



Is there a field gradient at a nucleus in a molecule? If the 
surrounding electrons are in an s-orbital, or constitute a 
closed shell, there is no field gradient. If however the electrons 
do not form a closed shell and are not s-electrons there may be 
a very strong field gradient (see Fig. H19 and Fig. Q2 on 
p. 187). Therefore the magnitude of the electric quadrupole 
interaction in a spectrum gives information about the state of 
"hybridization of the orbitals occupied by the electrons sur- 
rounding it (that is, the amount of p-orbital character in the 
surrounding electrons). There are, of course, complications. 
One especially important nuisance is that the valence electrons 
may distort the underlying closed shells. This distortion gives 
rise to a field gradient at the nucleus and therefore increases or 
decreases the strength of the quadrupole coupling: these are 
the Sternheimer anti-shielding and shielding effects. The dis- 
tribution of charges in the vicinity of the nucleus may also be 
estimated by pretending that the molecule consists of an array 
of point charges (see -dipole moment): ionic it y has been 
estimated in this fashion. 

Questions. What is meant by the term 'hyperfine interaction'? 
What two classes of interaction are there? What is the nature 
of the magnetic coupling between an electron spin and a 
nuclear spin? Why does this interaction disappear (when there 
is an external field present} when the electron is distributed in 
an s-orbital, or when it is in a p-orbttal but the molecule is 
rotating rapidly in a liquid? What information can be obtained 
from the magnitude of the anisotropic hyperfine interaction? 



Why does there remain an interaction in fluid solution? Of 
what is the Fermi contact interaction diagnostic? In the 
radical N0 2 the hyperfine interaction to the nitrogen nucleus 
is 55 G for the isotropic component and 13 G for the 
maximum of the anisotropic component: what is the s- and p- 
character of the nitrogen orbital contributing to the molecular 
orbital occupied by the unpaired electron? (Use Table 9.) 
What is the source of the electric hyperfine interaction? What 
nuclei may possess a quadrupole moment, and what does such 
possession signify? With what electronic property does the 
quadrupole interact? What information may be obtained from 
a study of the interaction? What is the complicating feature? 

Further information. See MQM Chapter 1 1 for a more detailed 
account of the magnetic hyperfine interactions. The way that 
they are employed in "electron spin resonance and "nuclear 
magnetic resonance (especially for -spin-spin coupling 
constants) is described by McLauchlan in Magnetic resonance 
(OCS 1). See also Lynden-fiell and Harris (1969), Carrington 
and McLachlan (1967), Atkins and Symons (1967), Ayscough 
(1967), Bolton and Wertz (1972), Slichter (1963), and 
Abragam and Bleaney (1970). The role of hyperfine inter- 
actions in spectra, and the structural information they may 
be used to obtain, are described in Sugden and Kenney 
(1965), Herzberg (1950), and Townes and Schawl ow 
(1955). See Freeman and Frankel (1967) for a basic dis- 
cussion. Magnetic and electric properties of nuclei are listed 
in Table 17. 



I 



ionization potential. The ionization potential (IP) is the en- 
ergy required to remove to infinity an electron from the orbital it 
occupies in an atom or molecule. The more tightty bound the 
electron the greater its ionization potential. The energy is 
normally expressed in electronvolts (1 eV is equivalent to 
96'49 kj mof 1 ), and its value depends on the orbital involved 
and the state of ionization of the atom. The first IP is the 
energy required to remove the least tightly bound electron 
from the neutral atom, the second IP is the energy required to 
remove the least tightly bound electron from the singly 
charged ion, and so on. IPs are a good guide to the chemical 
properties of elements, and the periodic table can be discussed 
in their light. A useful exercise of this nature is the discussion 
of the IP of the atoms from He to Ne along the first row of 
the periodic table (see Table 18). These can be rationalized in 
terms of "penetration and shielding effects. 

Questions. What is an ionization potential? The ionization 
potential of atomic sodium is 5-14 eV; how much heat is 
evolved when 1 mol of sodium atoms is formed from a gas of 
Na ions and electrons? Could an atom have a negative 
ionization potential? Account for the IPs of the elements of 
the first row in terms of penetration and shielding effects. 

Further information. Lists of ionization potentials will be 
found in Kaye and Laby (1956) and § 7b of Gray (1972). In 
Puddephatt's The periodic table of the elements (OCS 3) will 
be found a discussion of their dependence on penetration and 
shielding effects, and the role they play in determining 
chemical properties. This aspect is developed in detail by 
Phillips and Williams (1965). The ionization potential is used 



to calculate "electronegativities, and is a basis of the experi- 
mental technique of photoelectron spectroscopy (see 
"photoelectric effect). 

intermolecular forces: a Synopsis. The principal classifi- 
cation of the forces between molecules is according to their 
range. The short-range forces are repulsive, and reflect the 
increase in energy that occurs when two electron clouds are 
forced into contact and penetration. As two closed-shell 
species approach each other some of the electrons may be able 
to adjust their distribution so that they occupy energetically 
favourable regions, but the "Pauli principle does not permit all 
the electrons to settle into these regions, and the remainder are 
forced to occupy regions that tend to raise the energy of the 
approaching pair (they enter the "antibonding orbitals in the 
"molecular-orbital theory). This disruptive effect overcomes 
the attractive effect of the more favourably sited electrons, 
and so the energy of the pair of species rises sharply as the 
distance diminishes: this may be interpreted as a force that 
drives the molecules apart, and so we see the repulsive force to 
be rooted in the operation of the Pauli principle. 

Long-range attractive forces between molecules must exist, 
for otherwise no condensed phases would exist. The simplest 
attractive interactions are between charged species (ion-ion 
interactions, the tonic "bond), where the Coulomb force draws 
the components together until repulsive forces supervene. 
Most molecules are uncharged, but attractive forces still 
operate. In polar molecules these can be identified with the 
attractive interaction between the permanent electric moments 
of the molecules (the 'dipole-dipole interaction, or the dipole— 



115 



116 



intermolecular forces: a synopsis 



( » R < a 

\ R > 0, 



BOX 10: Intermolecular potentials 

Rigid spheres: 

VI 

Point centres: 

ViR) = DR- 6 
9 < 8 < 15 usually. Maxwell tan molecules have 5 = 4. 
Square well: 

!°° R < O 
-e o < R < ra 
R > ra. 
Sutherland: 

v *H-cm r>o. 

Lennard-Jones: 

Vift\ = DR~* - CR^ 
special case: (6, 12)-potentia1 

Buckingham : 

V{R) = 5e _dfl - CR^ - C'R~ % . 

Modified Buckingham (6-exp): 

[e/(1--6/a]] [ (6/a)exp(o-a/?/ff )-R*JR 6 1 

R <R ™ 

R is the value of R for which the upper expression for 

m 

ViR) attains a maximum. 
Stockmayer: 

V[R) = V LJ iR) + S.iR,Huth). 

where V AR) is the Lennard- Jones potential, and 

£(R, fi lr fa) is the dipole-dipole interaction energy given in 

Box 5. 



ViR) 



-{' 



point-charge interaction if one is charged, or the dipole— 
induced-dipole interaction if one is polar and the other not: in 
the last case the strength of interaction depends on the 
"polarizability of the nonpolar molecule). Then there are the 
forces between nonpolar molecules: the most important of 
these is the London "dispersion force which isal so termed an 
induced-dipole— induced-dipole interaction because it depends 
on the fluctuation of the electron density of one molecule 
leading to an instantaneous dipole moment which may in turn 
induce an instantaneous dipole moment in the other, these 
two dipoles then sticking together. This interaction depends on 
the polarizability of both molecules and its energy depends on 
their separation as R' 6 . 

The name van der Waals forces is a general term applied to 
these intermolecular interactions. The term Keesom force is 
reserved for the interaction between polar molecules, and 
London force refers specifically to the dispersion force. 

The distance-dependence of the forces and the angular 
dependence of those between non-spherical species are 
normally expressed in terms of an empirical formula which 
has more-or-less the correct qualitative form and contains only 
a few adjustable parameters. Some of the more common inter- 
molecular potentials are illustrated in Fig. 11 and Box 10. 
All make a crude approximation to the repulsive interaction 
which, in reality, because it depends on details of molecular 
■wavef unctions and -overlap, has a complicated behaviour. 
The most cavalier approximation is to replace the repulsive 
part by an impenetrable hard sphere. The Lennard-Jones 
potential assumes that the potential rises sharply at small 
separations according to R' n ; when, for mathematical con- 
venience, n is chosen to be 1 2, we have the {6, 1 2}-Lennard- 
Jones potential (the R~ 6 -dependence [see Box 10] reflectsthe 
presence of the dispersion forces). The Buckingham exp-6 
potential retains the R' 6 component but pretends that the 
repulsive forces vary exponentially. The Keesom potential is 
an expression for the interaction of non-spherical molecules 
which treats them as cylinders capped by hemispheres. The 
Stockmayer potential adds to the Lennard-Jones potential a 
term representing the interaction of two dipoles placed at the 
centres of the molecules. 

Intermolecular forces must be taken into account if the 
properties of real gases and liquids are to be understood, and if 



inversion doubling 



117 



rigid spheres 



poinf centres 





_o; 



Sutherland 



Buckingham 




FIG, II. The shape of some empirical iniermolecular potentials. 



quantitative calculations are to be made. One important route 
to thermodynamic quantities is through the virial equation of 
state, where PV is expressed as an expansion in 1/1/; the coef- 
ficents in this expansion, the virial expansion, are the virial 
coefficients. Expressions for these may be obtained from the 
intermolecular potentials and the coefficients used to calculate 
the properties of the gas. The structure of liquids depends on 
the form of the intermolecular forces. The forces {and the 
parameters in the empirical expressions) may be determined 
by fitting calculated virial coefficients to experimentally 
determined values, from transport properties, and, best of all, 
from scattering experiments In molecular beams, where 
individual molecules are used both as targets and projectiles 
and their deflexion is determined by their interaction. 

Further information. An excellent book on the subject, which 
has a synoptic review in an early chapter and detailed accounts 
in later chapters, is that by H irsc hf elder, Curtiss, and Bird 
(1954). A helpful account of Debye's contribution has been 
given by Chu (1967), and details of the quantum-mechanical 
equations involved, and their derivation, will be found in 
MQM Chapter 11, Kauzmann (1957), and Margenau and 
Kestner (1969). The determination of intermolecular poten- 
tials is described by Hirschf elder, Curtiss, and Bird (1954), 
Curtiss (1967), and Dymond and Smith (1969). Molecular 
beams and their application to the determination of inter- 
molecular potentials are described by Levtne and Bernstein 
{1974) and Ramsey {1956}, and reviewed in a collection of 
articles edited by Ross (1966). Experimental and theoretical 
aspects of intermolecular forces have also been reviewed by 
Hirschfelder (1967). 

inversion doubling. Inversion "doubling can be discussed 
in terms of the specific and important example of the 
ammonia molecule. This pyramidal molecule can vibrate in a 
symmetrical bending mode, rather like an umbrella being 
shaken dry. And, like an umbrella, there is some probability 
that it can be inverted into another configuration where it 
continues to vibrate. The original and inverted configurations 
are physically indistinguishable, and vibrate with the same 
frequency; this situation can be pictured as in Fig. 12, where 
the molecule vibrates in one of the two potential wells. But 



118 



inversion doubling 




frequency 






FIG. 12. Inversion doubling in ammonia: black lines are the 'confined' 
states of the two wells; coloured tines are the true, interacting states. 

the fact that the molecule has some chance of inverting, or 
of "tunnelling from one well into the other, implies that the 
"vibrational wavefunctions of one well (which represent the 
displacement of the nuclei) seep through into the other. If 
the vibrational wavefunctions were wholly confined to one 
well the molecule would not be able to invert. Since the 
wavefunctions seep through the potential barrier their 
amplitudes overlap, and, just as in the case of the formation 
of a -molecular orbital, the correct description of the 
vibrational state of the molecule is in terms of a -super- 
position of the functions. Thus we take either their sum or 
their difference. The two new functions so formed correspond 
to different energies, because one is concentrated more with- 
in the barrier than the other. It is important not to conclude 
that the lower state is the one with the node within the 
barrier, even though this has the lower potential energy. We 
must also, as always, take into account the -kinetic energy, 
and discuss the total energy. When a -node is introduced into 



a function it becomes more curved, and its kinetic energy 
rises. This is the case for the tunnelling states of ammonia, and 
an analysis shows that the kinetic energy dominates and that 
the unbuckled, nodeless, symmetrical function corresponds 
to the lower total energy. In this connexion see -torsional 
barrier. It follows that, instead of having pairs of "degenerate 
vibrational states in the ammonia molecule, the inversion 
causes the degeneracy to be removed and each pair splits into 
two: each level is "doubled. 

The energy separation depends on the difference in 
amplitude (and therefore probability) that the nuclei will be 
found within the potential barrier: this depends on the 
strength of the overlap between the vibrational wavefunctions 
in the two wells, and this in turn depends both on the height 
of the potential barrier above the interacting levels (the higher 
it is the smaller the penetration) and on the mass of the 
"tunnelling species (the greater the mass the less the tunnel- 
ling). In NH 3 , the splitting, which is known as the inversion 
frequency, is 23 786 MHz (0-793 cm -1 ) for the lowest level. 
On a classical picture of the process this frequency would be 
identified with the frequency with which the inversion 
actually occurred. In quantum mechanics that concept is 
untenable in detail, but a "wave packet localized in one well 
(and representing one configuration of the molecule) would 
wriggle through the barrier and emerge on the other side, with 
more or less the observed 'inversion frequency'. 

Questions. 1. Why is inversion doubling so called? What is its 
source? Why do the positive and negative combinations of the 
vibrational levels correspond to different energies? Which has 
the lower energy? What determines the separation? Discuss the 
dependence of the separation as the barrier height is reduced 
from infinity to zero. How does inversion doubling affect the 
spectrum of ammonia? What changes would you expect to 
observe on deuteration of ammonia? Discuss inversion 
doubling in the other Group V hydrides. 
2. Consider a double square-well potential with a rectangular 
potential barrier; take the potential V to be infinite for 
x<0and* >£, to have the values between x= ^L — b and 
jt +6, and to be zeroelsewhere.Set up and discuss the solutions 
of the "Schrb'dinger equation for a particle of mass m in this 
system, and discuss the doubling of the energy levels that is 



brought about by lowering the height of the barrier from 
infinity. Discuss the effect of changing the mass of the 
particle. 

Further information . See MQM Chapter 10. For the spectro- 
scopic consequences of inversion doubling see §9.19 of King 



inversion doubling 



119 



(1964), §8.2 of Sugden and Kenney (1965), various parts of 
Herzberg (1966), and Chapter 12 of Townes and Schawlow 
(1955). The inversion process is the basis of maser action (a 
microwave iaser process) and so has been studied in consider- 
able detail: for applications of this nature see the references 
under "laser, Vuylsteke (1960), and Troup (1963). 



J 



J. The letter J is worked hard in quantum theory, but the con- 
text normally eliminates confusion. J and/ are 'quantum 
numbers used to denote the total "angular momentum of a 
system: the former is used in a many-electron system, or for 
the overall "rotation of motecufes, and the latter is used when 
only a single particle is involved. As a consequence of this use 
both J and/ are used to distinguish the "levels of 'terms. J is 
also used to denote the ^spin-spin coupling constant in °n.m.r. 
and the 'Coulomb integral for the electrostatic interaction of 
electrons. J or j is used for the current, be it the current of 
electrons induced by applied fields in metals, atoms, and 
molecules, or the flow of matter, heat, and entropy. 

Jahn-Telter effect. To those for whom the natural tendency 
of Nature is to states of highest symmetry, the Jahn-Teller 
theorem is a bitter pill, for itscontention is that in a variety of 
situations a molecule of high symmetry is intrinsically unstable 
and will attain a lower energy by distorting into a lower 
symmetry configuration. To be more precise, the theorem 
states that no non-linear molecule can be stable in a 
"degenerate electronic state. Therefore, if the molecule is 
found, by calculation, to be degenerate, that form of the 
molecule will not be the stable form in Nature, and a less 
symmetrical and non-degenerate form will be its natural state. 
The theorem does not apply to linear molecules, and so these 
may exist in an undistorted degenerate state; but there is an 
analogous higher-order effect, the 'Renner-Teller effect, which 
takes care of that loophole. 

As an example of the Jahn-Teller effect we shall consider 
the case of a Cu 2+ ion in an octahedral environment. We 



proceed to deduce its electronic structure on the basis of 

-crystal -fie Id theory (which is good enough for our present 

purpose), and discover that its configuration ist 6 e 3 . This is a 

degenerate configuration because the two configurations 

d 2d 2 2 and d 2 2d 2 have the same energy. The 
2 x -v x -y z 3 ' 

molecule is non-linear, and so the Jahn-Teller theorem 
predicts that the molecule must distort and eliminate its 
degeneracy. 

The physical reason for this may be understood by con- 
sidering the forces operating on the iigands: in the first 
configuration there is more electron density along thez-axis 
than in the equatorial plane, whereas the opposite is true in 
the second configuration. In the first case there is a tendency 
for the Iigands along the z-axis to move away from the central 
ion, and for the equatorial Iigands to move in; in the second 
configuration the opposite shifts are to be expected. If the 
molecule does distort by stretching along the z-axis, the 
electronic configuration d* 3 d x 2_ 2 will have a lower energy 
than the other electronic configuration in the same environ- 
ment {Fig. J1 a). If the complex flattens along the £-axis, for 
that conformation the second configuration will have the 
lower energy (Fig. J T b). Therefore we see that there does 
indeed exist a distortion of the molecule that removes the 
degeneracy of the electronic states, and we should expect the 
molecule to be found in either the elongated or the compressed 
form. (Which form it attains is very difficult to predict.) 

Several electronic configurations are predicted to produce 
Jahn-Teller distortions in octahedral complexes: we simply 
have to seek cases in which degenerate configurations can arise. 



120 



Jahn-Teller effect 



121 




FIG, J1. The effect of (a] elongation and (bl flattening an octahedron 

on the energy of electrons in d 2 and d 2 2. 
z x —y 

The high-spin d 4 configurations t 3 e l , the low-spin d 7 con- 
figurations tV, and the d 9 configurations tV are all cases in 
which there is degeneracy because a single electron or a single 
hole has to occupy the two degenerate e-orbitals (d % and 
^22)- Examples are Cu 2+ (d 9 ) and Cr 2 * (d*), whose com- 
pounds often have closely related distorted structures, Mn J+ 
(d 4 ), and the rare low-spin Co 2+ (d 7 ), low-spin Ni 3+ (d 7 ) r and 
Ag 2+ (d 9 ). Jahn-Teller distortions are not expected for d 3 , 
high-spin d 5 , low-spin d* , or d 8 . 

Jahn-Teller distortions might be predicted also for con- 
figurations that give rise to degeneracy in the t-orbitals, such 
as d l , d 1 , high-spin d 6 , and high-spin d 7 . In practice the 
t-orbitals are directed between (rather than along) the metal- 
ligand axes, and uneven electron occupation therefore fails 
to produce observable distortions. 

The d 8 configuration is rather special in the sense 

that small distortions are not expected; if a distortion does 

occur it is large. Because the spins are parallel, one electron 

must occupy d 2 and the other 1 must occupy d 2 2 ;there- 
z x — y 



fore in a small distortion the energy of one electron increases, 

the other falls, and so overall there is no change in the energy. 
If the distortion is great enough, however, the energy differ- 
ence between the d 2 and d 2 2 orbitals can exceed the 
energy required to cram both electrons into either d 2 or 

d i 2 (that is, to overcome the first -Hund rule). Strictly 
x — y 

speaking, this is not a Jahn-Teller distortion because it does 
not lower the degeneracy of the system. 

Two effects complicate the analysis of the effect. The first 
is the dynamic Jahn-Teller effect, in which the centre of 
attention is the motion of a molecule in which a degeneracy 
may arise at certain nuclear conformations. The second is the 
role of °spin-orbit coupling, which generally reduces the 
effect's magnitude. The experimental detection of a true Jahn- 
Teller distortion is very difficult as there are other reasons why 
a complex may be distorted. In particular it is difficult to 
distinguish it from a distortion due to the packing requirements 
on formation of a crystal. 

Questions. What is the Jahn-Teller theorem? What molecules 
are excluded? When a molecule is predicted by calculation to 
have a degenerate electronic state, what are the consequences 
of the theorem? Account for the content of the theorem in 
terms of the crystal-field model of transit ion-meta! complexes: 
why may a lower energy be attained when an octahedral com- 
plex is distorted? What numbers of electrons must such a com- 
plex possess in order for this description to apply? Why is the 
Jahn-Teller effect much less important in tetrahedra! 
complexes? What situation arises when the configuration of 
the central ion is d 8 ? What are the methods for distinguishing 
the Jahn-Teller effect, and what are the complications? When 
is the dynamic Jahn-Teller effect important? What is the 
'ligand-field (MO) explanation of the Jahn-Teller effect? 

Further information. See MQM Chapter 10. A simple account 
of the Jahn-Teller effect is given in §4.2 of Orgel (1960) and 
Coulson (1961). Some of the mathematics is put in a simple 
way in §8d of Ballhausen (1962), and a full and interesting 
discussion of the spectroscopic consequences is given in §1.2 
of Herzberg (1966). A discussion of both the static and 
dynamic effects is given in a book devoted to the subject by 
Engl man (1972), and a simple example is worked on 
p. 45 of Herzberg (1966) and on p. 194 of Ballhausen (1962). 



K 



kinetic energy. The kinetic energy of a particle is the energy 
it possesses by virtue of its motion. In quantum mechanics the 
kinetic energy is related to the curvature of the -wavefunction. 
As the wavefunction becomes more sharply curved, so the 
kinetic energy of the state it represents becomes greater. Con- 
versely, a state represented by an almost flat function has 
virtually no kinetic energy. 

In the case of a free particle the wavefunction is of infinite 
extent and has a wavelength that decreases as the kinetic 
energy increases. The reason for this can be grasped quite 
easily from the connexion between curvature and energy: as 
the curvature of the wavefunction increases it becomes more 
buckled, and swings more rapidly from positive amplitude to 
negative; but increasing the rapidity with which the function 
changes sign is simply another way of saying the wavelength 
shortens. This connexion of wavelength and kinetiG energy, 
and thence momentum, of free particles is the explanation of 
the -de Broglie relation. 

In cases where the particfe is bound to a potential centre the 
wave may not be sufficiently extensive for a wavelength to be 
discernible or meaningful, but the relation between curvature 
and kinetic energy remains. An example of this situation is the 
"hydrogen atom: the ground state arises from the balance of 
the kinetic and potential energies; the 1s-orbital is a simple 
exponentially decaying function which never passes through 
zero, yet an electron in it possesses kinetic energy by virtue of 
the non-zero mean curvature of the exponential function. 

Questions. 1. What properties of the wavefunction determine 
the kinetic energy of a system? What is the connexion between 



the wavelength of a free particle and its kinetic energy? How 
is the wavelength related to the momentum? Calculate the 
wavelength of an electron with an energy equivalent to 1 eV, 
1 keV, 1 MeV {eV is "electronvolt). Discuss the role that the 
curvature of the wavefunction plays in determining the 
structure of the ground state of the -hydrogen atom; what 
would happen to the energy of the atom if the electron were 
pinched more closely towards the vicinity of the nucleus in 
order to lower its potential energy? 

2. The kinetic energy of a one-dimensional system is calculated 
by evaluating the -expectation value of the "operator— {b 2 /2rn) 
(d /dx }, and the corresponding operator for a three dimensional 
system is -{h 2 /2m) {9 2 /3x 2 + d 2 /V + d 2 /dz 2 ), or 
—{\i 2 /2m)V 2 in terms of the "laplacian. Evaluate the kinetic 
energy of a particle with the following wavef unctions: 
explkx, s'mkx, stnnkx s\r\mky, exp(- kx 2 ), exp(— nr/a ). 



Plot the kinetic energy of a particle with the last wavefunction 
as a function of n, and so see the connexion between the 
curvature of a hydrogen-like exponential function and the 
kinetic energy. 

Further information. See MQM Chapters 3 and 4 for some 
more specific examples. All standard books on quantum 
theory bring out the connexion of the curvature and the 
kinetic energy: therefore see Chapter 5 of Davydov (1965), 
Landau and Lifshitz U958a), and Messiah {1961). An 
interesting account that emphasizes the optical-mechanical 
analogy is given in Chapter 3 of Bohm (1951), Scattering 
phenomena are excellent examples of the wavelength's 
connexion with kinetic energy; therefore see books on 



122 



Koopmans' theorem 



123 



scattering theory, such as Rodberg and Thaler {1967) and 
Goldberger and Watson (1964), as well as the appropriate 
sections in the other books referred to above. Levine (1969) 
applies scattering theory to the discussion of molecular 
reactions; and this is presented in a more pictorial form by 
Levine and Bernstein (1974), 

Koopmans' theorem. 'Theorem' is a name too grand for 
this approximation; Koopmans' rule would be better, for it is 
transgressed frequently, and is never obeyed to the letter. The 
rule states that the 'ionization potential of an electron is equal 
to the energy of the orbital from whence it came. This appar- 
ently trivial statement is based on the fact that the energy that 
has to be supplied in order to ionize an atom or molecule must 
be expended in overcoming the combined effect of nuclear 
attraction energy and the electron's repulsive interaction with 
the other electrons present, and these factors determine its 
energy in its orbital. 

The rule is an approximation because it assumes that the 
remaining electrons will not reorganize themselves in order to 
take advantage of the absence of the electron which is being 
removed. Therefore, when one-electron energies are calculated 
in the Hartree-Fock "self-consistent field manner, the ion- 
ization potentials calculated by Koopmans' rule are often in 



error because it is assumed that the electrons in the ion occupy 
the same orbitals as they did in the atom. Furthermore, the 
Hartree-Fock scheme neglects electron "correlation effects, 
and is non-relativistic: the latter approximation can be 
seriously in error for electrons that are strongly bound and so 
subject to strong forces. 

Questions. 1. State Koopmans' rule. What is its justification? 
Why is it only an approximation? Would you expect it to 
overestimate or underestimate ionization potentials? For 
which electrons should it be a better approximation? For 
what species is the rule exact? 

2. The one-electron energies of the CO molecule are as follows: 
4a, 21 '87 eV; 50, 15- 09 eV; 17T, 17-40 eV. Estimate the ion- 
ization potential for the molecule when an electron is removed 
from these orbitals (experimental values are 19-72 eV, 14-01 
eV, 16-91 eV). Discuss the structure of the molecule in the 
light of these values (which are taken from p. 36 of Richards 
and Horsley, loc. cit. infra). 

Further information. Koopmans' rule is now of interest 
because of the development of photoelectron spectroscopy: 
for an account of this see Turner, Baker, Baker, and Brundle 
(1970), Molecular energies are listed in the bibliography com- 
piled by Richards and Horsley (1970). 



L 



laplacian. Pierre Simon de Laplace (1749-1827) was a 
notable French mathematician, and to his work his formidable 
Mecanique celeste is a profound and worthy memorial. 
Among its many notable pages of mathematics is the Laplace 
equation, ® 2 flbx 2 \ + (b 2 f/by 2 ) + (b 2 f/bz 2 ) = 0, where f is 
some function. Although the equation was set up in order to 
account for the properties of gravitational fields it has turned 
out to be applicable to a wide variety of phenomena. That it 
governs properties like the flow of incompressible fluids, 



BOX 11: The laplacian and the legendrian 
Cartesian form (x, y, z as coordinates} 

laplacian: *-& + £+£. 
Spherical polar form {r, 8, as coordinates) 



laplacian; V 2 = 



t(&)^ 



A? 



or V' 



2 0_ 

r br 



_ + i. 2. + J_ a 2 






b , /; 9 

0b6 smu ae 



or A : 



sinO 90 ' 30 2 



. cosfJ b_ , 



Cylindrical polar form (r, Q, z as coordinates) 



laplacian: V 2 = --f- + 



+ ^ 



1 d 2 



r or br 2 r 2 b6 2 bz 2 



gravitational and electromagnetic fields, and heat flow is 
probably due to the fact that it is an equation that expresses 
the tendency of natural phenomena towards uniformity: to 
the elimination of curvature in the distributions. So important 
is it that the differential operator {d 2 ibx 2 ) + (3 2 /by 2 ) + 
(b 2 !bz 2 ) is given a special symbol V 2 the laplacian, and read 
'del -squared'. Laplace's equation then becomes simply v*V= 0. 
It is not always convenient, and often foolish, to work with 
the laplacian expressed in cartesian coordinates; for systems of 
a predominantly spherical nature the spherical polar form is 
more convenient, and both forms are collected in Box 11. The 
part of the laplacian involving the angular derivatives is referred 
to as the legendrian and written A 2 . 

Further information . The laplacian occurs widely in quantum 
mechanics because it is the "operator for the "kinetic energy; 
it therefore appears as a component of the "hamiltonian and is 
the differential part of the °Schrddinger equation. For an 
account of its transformation from cartesian coordinates to 
spherical polars see §6.8 of Kyrala (1967) for swift methods 
and Appendix 5 of Moelwyn-Hughes (1961) for slow. A brief 
biography of Laplace has been written by Newman (1954). 

laser. The word laser is an acronym formed from 'light 
amplification by stimulated emission of radiation' and is a 
development of the maser, where the m denotes microwave, 
according to some, or molecular according to others. A laser 
operates by absorbing energy and emitting it at a well-defined 
wavelength by a stimulated emission process (see "Einstein A 
and fi coefficients). As a simple example, consider a sample of 



124 



material in which most of the atoms are in an excited state, 
and let the sample be contained in a cavity with reflecting 
walls. One atom will emit a photon as it falls spontaneously 
into the ground state, and this photon will rattle around inside 
the cavity, its presence stimulates another atom to emit, and 
so a second photon appears in the cavity and travels in phase 
with the first. The pair of photons stimulates another 
emission, and very rapidly a cascade of emission occurs and an 
intense light field grows inside the cavity. The process ceases 
when the population of the excited state has fallen to its 
equilibrium value. If one of the walls of the cavity is semi- 
transparent, light will leak through it and an intense, mono- 
chromatic burst of phase-coherent light will emerge. 

How is the process realized in practice? The cavity is 
normally in the form of a long thin tube, if the sample is a gas, 
or a cylinder of material if it is a crystalline solid; this configur- 
ation implies that the emitted radiation will have a very small 
lateral divergence because only those photons ricocheting 
backwards and forwards along or very close to the axis con- 
tribute to the amplification cascade: the remainder pass 
through the side walls and drain away from the system before 
their intensity amplifies. Thus the geometry of the system 
leads to a beam with very little divergence. 

The principal problem that remains is how the population- 
inversion necessary for stimulated emission, and therefore for 



IB 







purrp 



c=& 



light 



pumpCr^ 



IITW1 1 



JiaSt 



heat 



liWlT 



'radiotionless 
I decay 



FIG. L1. (a) A iwo-leve! laser and (b) a three-level laser. 



125 

laser action, is achieved. Two basic systems may be envisaged. 
In the first (Fig. L1 a) the laser action takes place between an 
excited state and the ground state: the difficulty of this 
primitive arrangement is that it is necessary to remove more 
than 50 per cent of the atoms from their ground state into the 
excited state, The second method obtains laser action between 
two excited levels: the population-inversion is much easier to 
attain, especially if the lower excited state can relax rapidly 
into the ground state. In such a three-level laser {see Fig. LI b) 
intense radiation excites the absorbing atoms or ions into the 
uppermost level, and if this pumping step is sufficiently 
effective the population in that level {E 2 ) is significantly, and 
even enormously, greater than that in the lower excited level 
E t . If the E 2 population does not leak away into the lower 
levels by non-radiative processes, laser action may take place 
between E 2 and E , because photons passing back and forth 
within a reflecting cavity stimulate the B%~ *fi emission; a 
leaky mirror at one end of the cavity permits a highly 
collimated {non-divergent), coherent (all waves in phase, 
because they are generated by stimulation), polarized (because 
of the polarization of the transition or because the cavity has 
polarizing windows}, and monochromatic radiation of 
frequency P= (f 2 - E h )lh to emerge. 

An example of the laser process is the helium-neon laser in 
which the active material is a mixture of the two gases in the 
ratio 1 :5. A radio discharge, being no respecter of "selection 
rules, excites the helium atom, and although many of the 
excited states decay very rapidly, the first excited singlet state 
is relatively long-lived (because s-s transitions are forbidden), 
and atoms in this state swim around in the sample for some 
time because they cannot return to the ground state except by 
a forbidden transition. Whilst swimming an excited helium 
atom may collide with a neon atom in its ground state, and by 
a coincidence (a coincidence at the heart of this laser system) 
the energy that the helium atom possesses almost exactly 
matches the energy of an excited state of neon. There occurs 
-resonant transfer of energy, the helium is deactivated and the 
neon is in an excited state E 3 (Fig. L2). This state has two 
unpopulated energy states beiow it which do not resonate 
with excited helium. Therefore it is in a configuration typical 
of a three-level system; laser action occurs between E 3 and E\, 
and red light is emitted at 632'82 nm. 



126 



laser 
He 



r 



2'S 



4^y === ^ 



2h- 



discrieitje ■ — / 



ft. 



collision 




FIG, L2. The Me— Ne laser (for the 632-8 nm mode). 

Ruby is a famous laser, and is an example of a sort of three- 
level system involving four levels, but acting like a two-level 
system in so far as laser action involves a transition to the 
ground state. These somewhat perplexing remarks can be 
resolved by a glance at Fig. L3. Ruby, which the prosaic 
know to be Al 2 3 with about 5 per cent of the Al 3+ ions 



heat 



heat 




pump 



6943 ..... 



FIG. L3. The ruby laser transitions. 



replaced by Cr 3 *, gives delight to poets by virtue of its two 
intense transitions in the green (to E 2 ) and the violet (to E 3 ). 
Having pleased poets, ruby proceeds to please physicists as its 
chromium ions drop from E 3 and E t into E t : this is a 
radiation/ess decay and relies on the transfer of the electronic 
excitation energy into the vibrations of the surrounding lattice 
(which gets hot). The initial absorption is so efficient that 
when intense illumination is used the population of E t may 
exceed that of the ground state, and laser action occurs with 
the emission of red light {694'3 nm). 

Lasers may be either continuous wave (CW), when the light 
emerges in a continuous flow so long as the pumping 
operation is in progress, or pulsed, as in the case of ruby, when 
the laser light emerges in a short burst as the stimulated 
emission occurs. The emergence of the light in short bursts 
means that very high powers may be obtained, albeit only for 
very short times. A typical procedure in a ruby laser is to 
employ a brilliant pumping flash emitting about 2-5 kJ of 
energy. Much of this is absorbed by the ruby rod, and although 
much of the absorbed energy appears as heat, about 25 J 
appears as laser radiation. But the pulse of light lasts for only 
about 5 X 1 CT's, and so the power that emerges is a splendid 
50 kW, but of course this power is maintained for only 
5X10 s. It is possible to increase the power output by 
shortening the length of the pulse in which the same amount 
of energy is delivered: the technique of effecting this is 
Q-switching. A crude way of describing this is to imagine a 
cavity with one of the mirrors removed; then pumping 
radiation is applied in a flash and the population of the upper 
level attains its maximum value and is not depleted by 
stimulated emission. Then the lost mirror is hurriedly slapped 
back into place while the population is still inverted, and it 
all drops out, by stimulation, and a giant pulse is obtained in 
about 10~ 8 s. The power of the laser, if the pulse carries 25 J, 
is a massive 2-5 X 10 9 W, which is roughly the output of a 
massive power station (but power stations have the 
economical advantage of working for some years at that level, 
rather than for 10~ 8 s,. The removal and replacing of the 
mirrors was done mechanically in early models but now the 
Kerr cell or Pockels cell is an electrical substitute. 

Some of the common laser materials and their wavelengths 
are listed in Table 19. 



127 



Further information. See MQM Chapter 10 for a further dis- 
cussion of the ruby system, A simple account of lasers is given 
by Schawiow (1969), Wayne (1970), and Lengyet (1971), and 
in review articles by Jones (1969) and Haught (1968). The last 
gives information about the expressions mode-locking (a 
manner of achieving short, picosecond flashes), Q-switching, 
various laser systems, and many pertinent references. A simple 
account of liquid lasers has also been given by Heller (1967). 
The stimulating source need not be radiation: chemical re- 
actions that leave product molecules in excited states are also 
the basis of laser action— a chemical laser is a remarkable 
device that turns chemical energy directly into coherent light. 
For an account see Haught (1968) and Levine and Bernstein 
(1974), Chemical applications of lasers reviewed by Jones 
(1969) and Haught (1968). See Pressley (1971) for a com- 
pilation of data on lasers, and Levine and OeMaria (1 966 et 
seq.) for recent advances. 

level. In atomic structure and spectroscopy the name level 
denotes a particular value of the total 'angular momentum J. 
For example, from the "configuration 1s 2 2s 2 2p arises a 
doublet P term written J P. The two levels of this doublet 

3 1 

correspond to J = ^ and J = ^> * or only these a n 9ular 
momenta may be constructed from 1 = 1 and S = \ (see 
"angular momentum). The two levels of the term therefore are 
written 2 P, /2 and ^P^. 

The number of levels of a particular term is its 
"multiplicity. Thus in 2 P there are two levels, and in 3 there 
are three levels ( 3 D t , 3 D 2 , and ?D 3 ). The levels differ in energy 
because of the °spin-orbit coupling, and their order can be 
predicted on the basis of the -Hund rules. The angular 
momentum./ may have 2/ + 1 different values of its 
component on some arbitrary axis; in common with the 
practice of angular momenta these are distinguished by the 
value of Mj. The state of a level of a term is denoted 2 ? y J , 
etc. 

Questions. Define the use of the term 'level' in atomic 
spectroscopy. Write down the levels that may arise from the 
following terms: l P, 3 f, 3 S, 4 D, 6 D. Under what circumstances 
does the superscript not denote the mutiplicity? (See "term 
symbols.) 



Further information. See Chapter 8 of MQM for more detail 
about atomic spectra. Levels, multiplicity, and fine structure 
are discussed by Kuhn (1969) and Herzberg (1940). Turn to 
•atomic spectra and -fine structure for further information. 

ligand-field theory: a synopsis. The ligand-field theory of 
the structure of complexes of transition-metal ions is an 




s.Q) %o> 




FIG. L4. The 6 ligand orbitals and the 6 sy m me iry -adapted 
COmbi nations. 



128 



ligand-fiekl theory: a synopsis 



extension of the 'crystal -field theory which takes into 
account the known derealization of the electrons of the ion 
into the orbitals of the surrounding ligands; at the same time it 
makes use, like the crystal-field theory, of the very high degree 
of symmetry of the complexes normally encountered. The 
ligand-field theory is essentially a 'molecular-orbital theory of 
complexes, and begins by concentrating its attention on the 
d-electrons of the central ion. We shall illustrate the method by 
considering an octahedral complex in which the central ion 
possesses n d-electrons. 

Consider the ligands as bearers of (J-orbitals which approach 
the central ion with a -lone pair of electrons. We denote these 
by the spheres in Fig. L4. From these six orbitals six combin- 
ations may be formed: the six chosen have well-defined 
symmetry properties {For those who know "group theory, we 
select "linear combinations that span irreducible represen- 
tations of the octahedral point group.) The six combinations 
are illustrated in the figure: they fall into three groups. Only 
one of the groups, that labelled e, has a net "overlap with the 
d-orbitals of the central ion, and so only this combination can 
form bonding and "anti bonding molecular orbitals with the 
d-orbitals. It follows that a molecular-orbital diagram of the 
type shown in Fig. L5 may be anticipated. The energies of the 
ligand and ion orbitals are such that the lower-energy combin- 
ation is largely ligand in nature, and the antibonding combin- 
ation is nearly metal ion in character. 



liqandi 



odd n elections 



complex 




largely metal 
character 



i- largely ligand 
character 



FIG. L5. The figand-field splitting in an octahedral complsK. 



Into this set of eleven orbitals (a doubly "degenerate 
bonding-orbital labelled e , four degenerate nonbonding- 
orbitals confined entirely to the ligands and labelled t iu and 
a lg , three triply degenerate nonbonding-orbitafs confined 
entirely to the metal ion and labelled t^, and an antibonding 
combination labelled e*) we must insert [12 + r?) electrons 

g 

(2 from each ligand and n from the ion). Applying the 'aufbau 
principle it should be clear that the first 12 electrons will 
occupv the orbitals e , t , a , and that these are of predomi- 

r * g lu' jg' 

nantly ligand character. The next/i electrons have to compete 

for places in the orbitals t„ and e*: this is just the situation 

2g g 

encountered in the "crystal-field theory and Figs. C6 and C7 

should be consulted; there we see that if the energy gap 

between the orbitals is large {the strong-field case) all the 

electrons attempt to enter the t set, and enter the upper e 

set only if the "Pauli principle forbids them entry into the 

lower set. If the orbital separation {there called lODq) were 

small (the weak-field case) it might be energetically favourable 

for the electrons to enter the t and the e* orbitals, but to do 

2g g 

SO with their spins parallel ( "Hund rules). 

The distinction between the strong- and weak-field cases, 
and their generation of low- and high-spin complexes, is 
carried over from the crystal-field theory into the molecular- 
orbital theory: the difference lies in the source of the splitting. 

Another difference is the fact that the e*-orbital is not wholly 

g 
confined to the metal ion: since it is formed from the overlap 

of metal and ligand orbitals it contains some ligand character. 

This means that any electrons that occupy it may spread over 

on to the ligands. The evidence that this happens comes from 

spectroscopy, especially "electron spin resonance (where 

"hyperfine structure due to the ligands has been observed). 

The other important improvement of ligand-field theory 

over crystal-field theory is the natural way that the former 

permits it-bonding between the metal and the ligands. This 

is especially important when the ligands are species such as 

CO or NO. In order to see the effect of permitting Tr-bonding 

consider the octahedral complex again, and this time, in 

addition to the a-bonds, let each ligand possess two orbitals 

that are perpendicular to the rnetal-ligand bonds. From these 

12 orbitals, which are illustrated in Fig. L6, 12 combinations 

may be constructed. Three of the combinations have the same 

symmetry as the d 



xy 



d -,d -orbitals of the central ion 

xz yz 



linear combination of atomic orbital 5 ( l.CAO) 



129 




FIG. L6, The 1 2 jr-orbrtals of octahedrally disposal) Ngands. 




FIG. L7, ir-bonding: the effect on energies. In (a) the ligand 7r-levels 
are initially full; in (.b) they are empty. 



{which so far have formed the nonbonding t set of orbitals 
confined to the metal). When we permit this overlap to occur 
the molecular-orbital energy-level diagram is modified. Two 
situations need to be distinguished: in the first the TT-orbitals 
of the ligands are full (and lie below the tf-orbitals); in the 
second they are empty (and iie above the rj-orbitals). The two 
cases are illustrated separately in Fig, L7. In the former case 
the ligands bring up their 7T-e!ectrons and fill all the bonding 
combinations of Tr-orbita!s. This leaves the n electrons from the 
metal to be distributed between the antibonding it combin- 
ation (t* ) and the untouched e* combination. Since the 
previously nonbonding orbitals nave become slightly antibond- 
ing, the gap (10Dc/) has been reduced by the presence of the 
TT-electrons. The opposite is the case when the 7T-orbitals of the 
ligands are empty, for now the n electrons compete for places 

between the bondinq t combination and the antibonding e* 
29 9 

combination. Thus the splitting has been increased by the 

presence of unfilled ir-orbitals. 

The size of the splitting lODq determines the spectroscopic, 

magnetic, and chemical properties of the complex in the same 

way as there are determined by the "crystal -fie Id theory, but 

the distribution of the electrons on to the ligands gives them 

some of the latter's "spin-orbit coupling energy. 

Further information. See MQM Chapter 9 for an account of 
ligand-f ield theory. A simple introduction to the Ideas of 
ligand-field theory is provided by Earnshaw and Harrington in 
The chemistry of the transition elements (OCS 13); this is 
developed further in Coulson's The shape and structure of 
molecules (OCS 9), The effect of these ideas on the explan- 
ation of the magnetic properties of complexes is described by 
Earnshaw (1968). For spectral consequences see Jorgensen 
(1962, 1971). A simple and good introduction to ligand-field 
theory has been given by Orge! (1960), Coulson (1961), and 
Murrell, Kettle, and Tedder (1965), and developed more 
mathematically by Figgis (1966), Ballhausen (1962), and 
Griffith (1964). See also -crystal -field theory and the °Jahn- 
Teller effect. 

linear combination of atomic orbitals (LCAOh An 

LCAO is a method of describing a molecular orbital covering 
several nuclei in terms of a sum of atomic orbitals centred 



130 



linear combination of atomic orbitals (LCAO) 



on each nucleus. Thus the bonding orbital in the hydrogen 
molecule is really a complicated function spreading round 
both nuclei, but since it is expected to resemble the 1s-atomic 
orbitals on each nucleus the orbital is expressed as the LCAO 
formed by the "superposition of the two Is-orbitals (Fig. L8). 



that lies deeper than the mathematical device of being able to 
expand any function in terms of a sufficiently complete set of 
simpler functions. This deeper justification is provided by the 
"superposition principle. When one has the possibility that a 
variety of processes can occur the superposition principle 




IG. L8. In the LCAO method the wiavefunctions of the atoms (black) 
are superposed to give a molecular orbital (colour). 



If all the atomic orbitals were used in order to reconstruct 
the true molecular orbital the LCAO procedure would be 
exact, but this task is too heavy and normally a small number 
of atomic orbitals is selected as the basis set; used in this way 
the LCAO procedure is an approximation. In the elementary 
discussion of the "hydrogen molecule the basis set consists of 
the two 1s-atomic orbitals, and this gives a reasonably good 
description of the bond, but it can be improved considerably 
by expanding the basis to include the 2s- and 2p-orbitals, and 
others. The use of a small basis set in the LCAO description of 
molecular bonding is one of the gravest sources of error in the 
method, and the selection of too small a set can make a 
nonsense of an otherwise elaborate calculation. 

The use of the LCAO method has a theoretical justification 



demands that one should consider the probability amplitudes 
for the individual processes rather than the probabilities 
themselves. The total, composite process is described by the 
total, composite amplitude, and all manipulations and thought 
must be applied to this object, and the probability for an 
individual process extracted only at the end of the calculation. 
In the case of a molecule we have an example of such a 
composite situation, for there is a probability that an electron 
is on one of a number of nuclei. Our problem is the distribution 
of this electron in the face of these various possibilities, and 
the superposition principle tells us that in order to find this 
distribution we should construct the overall amplitude 
(molecular-orbital wavefunction) by superimposing the 
amplitudes of the individual processes (the atomic-orbital 



localized orbitals 



131 



wavef unctions). But this is precisely the line taken by the 
LCAO method in its attempt to construct the molecular 
orbital from the individual atomic orbitals. 

Questions. What is the LCAO procedure? Is it accurate? What 
is the worst source of error in the method? Can the method 
be justified? What is the basis set in the simplest description 
of molecular oxygen (that is, we require the minimal basis set)? 
How cart a simple description of molecular hydrogen be 
improved? What guidance can the physical nature of the 
problem provide in the problem of extending the minimal 
basis set? (Think about energy, size, and orientation of the 
orbitals, and how the polarization of an atom by neighbouring 
nuclei can be taken into account.) 

Further information. The LCAO method is at the foundations 
of molecular-bonding theory, and more information will be 
found under appropriate entries. See in particular Coulson's 
The shape and structure of molecules (OCS 9) and MQM 
Chapter 9. See also Coulson (1961), Murrell, Kettle, and 
Tedder (1965). and McGlynn, Vanquickenborne, Kinoshita, 
and Carroll (1972). For a simple introduction to °self- 
consistent field calculations on molecules see Richards and 
Horsley (1970). The choice of orbitals to combine into a 
particular molecular orbital must conform to the symmetry of 
the molecule, and "group theory can be extremely useful for 
determining appropriate combinations. See MQM Chapters 5 
and 9, Cotton (1963), Tinkham (1964), and Bishop (1973). 

localized orbitals. The chemistry of many molecules suggests 
that to a significant extent electrons may be regarded as 
belonging to different parts of the molecule; "molecular- 
orbital theory gives the impression of predicting that all 
electrons are spread throughout each molecule, and therefore 
it seems to run counter to the chemical evidence. The 
deficiency is apparent rather than real. It is possible to 
manipulate the form of molecular orbitals (by taking various 
sums of them) and to generate localized orbitals, orbitals that 
are localized almost wholly in the vicinity of different groups 
of the molecule. This procedure is illustrated in Fig. L9 (which 
is based on the calculation in Question 2), 

It is possible to take the appropriate linear combinations 
because the actual many-electron wavefunction of a molecule 



basis , 
orbitals 




locofe«i orbitals 



e 



V 





w 



& 



M&. 



fully localized orbitak 




>u=A 



FIG. L9. The formation of localized orbitals. Only when m 
localization obtained. 



X is full 



must satisfy the °Pauli principle, and this is ensured if it is 
written as a 'Slater determinant. It is an elementary feature of a 
determinant that rows or columns may be added to other 
rows or columns without changing its value. Therefore similar 
manipulations applied to the Slater determinant do not change 
the total wavefunction, and the localized description is 
mathematically equivalent to the original delocalized descrip- 
tion. An example of the procedure is set as a Question. 

Questions. 1. What is the aspect of the molecular-orbital theory 
that seems to conflict with chemical evidence? What evidence 
supports the view that electrons are indeed delocalized 
throughout the molecule? How are localized orbitals generated? 
What do they succeed in achieving? Why is it a permissible 
procedure? Why are delocalized and localized descriptions 
equivalent? Suppose we have an orbital which is a sum of an 
s-orbital on atom a, an s-orbital on atom b, and an s-orbital on 
atom c, and we call this \ii u then the orbital is tt. = c s + 

r ' a a 

c b s b C eV" and let there ^ a similar orbital in which the 



132 



lone pair 



central atom b contributes a p-orbital— then the bonding 
orbital is \Jj 2 = c s + c'a, + c s . Show that the combin- 
ations 0] + 2 and i^i — \p 2 are largely localized in the 
a— b and b— c regions respectively, and sketch the resulting 
localized orbitals. The reason why it is correct to take these 
combinations is treated in the next, harder Question, 
2. Consider a linear triatomic ABj molecule (BAB), and let the 
occupied orbitals be of the form ipt — s + M(a + b) and 
02 = P + ^fa — b) (for the notation see Fig. L9), These 
orbitals contain four electrons, and so the antisymmetrized 
wavefunction is the "Slater determinant ip= (1/4!)^ 
l^ iQ (1)0 (2(i// {3)0 (4) I, where a and denote the spins. 
Show that this determinant may be manipulated, without 
change of value, into the determinant ip = (1/4! ) Yl 
li/£(1) 0*(2) a (3)t//j(4) I, where 0*= (1)^(01 ± 2 ). Form 
these orbitals from the original pair and show that they are 
localized, but that the localization is complete only in the case 
where u = X. The procedure is illustrated in Fig. L9. Discuss 
your result in terms of the "hybridization of the central atom. 

Further information. See MQM Appendix 9.2 for an account of 
localization and a worked example. See also Coulson's The 
shape and structure of molecules (DCS 9) and Coulson (1961). 
See also Murrel!, Kettle, and Tedder (1965), Slreitweiser 
(1961), Salem (1966), and Pilar (1968). Good accounts in the 
literature on the formation of localized bonds are those of 
Lennard-Jones and Pople (1950), Boys (1960), and Edmiston 
and Ruedenberg (1963, 1965). 

lone pair. A lone pair of electrons is a pair of electrons of the 
valence shell not engaged in bonding. As an example, consider 
the tetrahedral distribution of electrons around the oxygen 



atom in HjO: two electrons stick one proton to the atom, two 
stick the other, and the remaining four form two lone pairs 
sticking out like rabbit's ears on either side of the molecular 
plane. We may consider the last four to be electrons in 
-localized, nonbonding orbitals. 

Lone pairs are important both structurally and chemically. 
They influence the structure of a molecule by exerting large 
repulsive effects on the electrons in neighbouring bonds: for 
example, the pyramidal shape of ammonia may be traced 
partly to the effect of the single lone pair on nitrogen 
exerting a repulsive force on the six electrons in the three 
N— H bonds. This analysis of molecular structure in terms of 
the lone-pair interactions is the basis of the Sidgwick-Powell 
rules which state that tone pairs dominate the repulsive inter- 
actions in molecules. In chemistry the lone pair is a nucleo- 
philic centre because it can so readily form a dative bond to an 
electropositive centre: lone pairs act as a base (in the Lewis 
sense). 

Since lone pairs are not tied into place by a parasitic 
nucleus they also contribute strongly to spectra, and the 
it** — n (read 'n to pi star') transition in carbonyl compounds 
is a major mode of excitation (see colour). The n stands lor a 
lone-pair (nonbonding) orbital, and the transition takes the 
electron from the oxygen lone pair and spreads it over the 
carbonyl group by depositing it in the -antibonding JT-orbital. 
Note that this transition changes the charge distribution in the 
carbonyl group and so is strongly responsive to solvent effects. 

Further information . See MQM Chapter 10. For a discussion of 
the Sidgwick-Powell rules and their modern development see 
Bader (1972). For references to the role of lone pairs in 
spectra see "colour. 



M 



magnetic dipole and electric quadrupole radiation. 

The most intense transitions in molecules are due to -electric 
dipole transitions, but when these cannot operate (when they 
are forbidden by 'selection rules) other mechanisms may have 
sufficient strength to cause a transition, albeit at a much lower 
intensity. One such mechanism is the magnetic dipole tran- 
sition, which relies for its operation on the magnetic com- 
ponent in the light field; the other mechanism, of similar 
strength, is the electric 'quadrupole transition, which relies 
for its operation on the variation of the electric field of the 
light over the space occupied by the molecule (that is, there 
must be a field gradient on a molecular scale if the quadrupole 
transition is to operate). 

The magnetic dipole transition generates the magnetic 
component of a light field just as an electric dipole transition 
generates an electric component {but, of course, in each case 
the other component is forced to accompany the generated 
component). There are two principal differences. 

The first is the weakness of the interaction of the molecule 
and field via the magnetic dipole. Pictorially, this can be traced 
to the rotational nature of the magnetic dipole transition: if 
during a transition charge is displaced in a curved path it wilt 
possess a magnetic transition dipole (Fig. M1). But in a region 
as small as the extent of a molecule the curvature of the dis- 
placement will be only weakly apparent: if D is the diameter 
of the molecule and Xthe wavelength of the emitted or 
absorbed light, it is plausibie to suppose that the efficiency of 
the coupling is of the order of DfK Since the intensity of a 
transition is proportional to the square of its "transition dipole 
moment, this suggests that the intensity should be only about 



{D/X) 2 of the intensity of an electric dipole transition, where 
curvature need not be detected. For typical molecules and 
wavelengths (D/X) 2 ~ 10~ s , and this is the order of magnitude 
of the intensities observed in practice. 

The other difference between electric and magnetic dipole 
transitions is in the selection rules: a magnetic dipole transition 
is akin to a rotational displacement of charge; a rotation is not 
reversed when it is inverted through a point (Fig. Ml); there- 
fore a magnetic dipole transition has even parity (see °gerade 
and ungerade\. Unlike an electric dipole transition (which is a 
translation of charge, and which is therefore of odd parity, 



electric dipole 
transition 




magnetic dipole 
transition 




\ I I J > J electric quadrupol 



transition 



FIG. M1 . The charge displacement in transitions of different type. 



133 



134 



magnetic moment 



and where the selection rules AL = ± 1 and g — ► u emerge) in 
a magnetic dipole transition AL = 0, g — * g, and u — ► u. The 
over-all selection rule AJ= ± 1 applies to both the electric and 
magnetic dipole transitions, because both are dipolar. (See 
'■electric dipole transition, and the account of the role of 
photon angular momentum.) 

The electric quadrupole transition arises from a displace- 
ment of charge that has a "quadrupolar nature. This somewhat 
subtle (but simple) type of charge displacement, which is illus- 
trated in Fig. Ml, can be detected with an efficiency of the 
order of D 2 1 A 2 (in intensity), and so we expect magnetic dipole 
and electric quadrupole transitions to be of comparable inten- 
sity. But detailed analysis shows that the latter also depends on 
the square of the frequency, and for visible light the intensity 
is reduced by a further two orders of magnitude. Therefore, an 
electric quadrupofe transition has an intensity of only 10" 7 that 
of an electric dipole transition. 

Since we are now dealing no longer with a dipole but with a 
quadrupole, the selection rules differ. A quadrupole can be 
envisaged as two dipoles in opposition: therefore we expect it 
to be of even parity. Consequently g — * g, u —* u transitions 
are allowed. More detailed analysis shows that the angular 
momentum selection rules are AL = 0, ±1 , ±2 and AJ = 0, ±1 , ±2. 
One might ask what has happened to our arguments concern- 
ing the unit spin of the 'photon? The answer lies in the quad- 
rupolar nature of the transition: as the photon is flung off the 
radiating molecule the spatial variation of the transition 
endows it with an orbital angular momentum. The total angular 
momentum of the photon may exceed unity, and so a selec- 
tion rule of AJ = ±2 can still be understood in terms of the 
conservation of angular momentum. 

Further information. Magnetic dipole transitions are nicely 
discussed in §IIID 2b of Kuhn (1962), in §3.2.2 of Griffith 
(1964), and in §7.6 of Hameka (1965). See also Heitler (1954) 
and Berestetskii, Lifshiu, and Pitaevskii (1971). The tran- 
sitions that are responsible for "electron spin resonance and 
"nuclear magnetic resonance, where a magnetic moment couples 
with an oscillating electromagnetic field, are important 
examples of magnetic dipole transitions. If a molecule can be 
excited to the same state by an electric and a magnetic dipole 
transition it is optically active (see "birefringence), 



magnetic moment. The magnetic moment of an electron due 
to its "orbital angular momentum (which may be pictured as 
arising from a circulating current) is u, = 7 I, where 1 is the 
orbital angular momentum and 7 n a constant of proportion- 
ality known either as the magnetogyric ratio or as the 
gyromagnetic ratio (the former name is more helpful). 
Simple calculation shows y s to be equal to — e/2m ; 
the negative sign of 7 (which arises from the negative charge 
of the electron) shows that the direction of u. is opposite to 
the direction of 1, but they are collinear (see Fig. M2). 




piofon 



, MC -9^ 



Af^/V** 



FIG. M2. Orbital and spin magnetic moments. 



The "spin angular momentum also gives rise to a magnetic 
moment (as a simple picture of a rotating charge would suggest) 
but its magnitude is 'anomalous' (which means that people 
could not explain it when it was first encountered), and in 
order to express the moment in terms of the magnetogyric 
ratio 7 an extra factor (the "g factor, is introduced: then 
jJ^ = gy s. Experiment, and later theory, showed thatff = 
2 , 0023, and this is frequently approximated as 5 = 2. The spin 
magnetic moment and the spin momentum are collinear but 
anti parallel. 

The magnetic moment is often expressed in terms of the 
Bohr magneton U = eh 12m , which is a positive quantity and 
may be considered to be a basic unit of magnetic moment (its 



magnitude is 9-273X 10 



J T 1 , or 9-273 X 10T 2 * JG"'). In 



-|U fh and so 



terms of the magneton we have 7 
M L = -(P B /h)l and n s - -"s(/u B /li)„. 

Just as the electron is the elementary negative charge and 
representative of the lighter fundamental particles (the 



magnetic properties 



135 



leptons, but note that the class lepton contains neutrinos and 

muons) and £l B is the elementary unit of magnetic moment, it is 

convenient to consider the proton as the elementary positive 

charge and representative of the heavier fundamental particles 

(the baryons, the proton and neutron, a subset of this class, 

are called nucleons). The corresponding elementary unit of 

magnetic moment is the nuclear magneton , /u N = eh/2m p , and 

it has the magnitude 5051 X ICT^JT'or 5-051 X 10" 3I JG -1 . 

The enormous difference in magnitude between the Bohr and 

nuclear magnetons is due to the difference in mass of the 

electron and the proton (to achieve the same spin angular 

momentum as an electron, the heavier proton needs a much 

smaller angular velocity, and so the equivalent current loop 

carries much less current, and the magnetic moment is smaller). 

The nuclear magneton is of a size convenient for the expression 

of the magnetic moments of nuclei (and, incidentally, of the 

magnetic moments of rotating molecules: see -^-value). If a 

nucleus has a spin angular momentum I its magnetic moment 

isff n (M N /h)l, where the nuclear g-value, which depends on the 

nucleus, is defined by this relation and is determined by 

experiment (and in the future, one hopes, by calculation in 

terms of the nuclear structure). It is found that g may be 

n 
positive or negative, depending on the element and the isotope, 

and it should be food for thought to be told that even the 

uncharged neutron has a magnetic moment. Typical values of 

the magnetic moments of some common nuclei are listed in 

Table 17. Notethat? =g (Ji„,/h). 

Further information. See MQM Chapters 8, 9, and 10 for a 
discussion of magnetic moments. The classical electromagnetic 
theory concerning magnetic moments is well described in 
Corson and Lorrain (1970). The Maxwell equations are sum- 
marized in Table 20. The quantum-mechanical theory of 
magnetic moments is described further under "spin, Rvalue, 
and "magnetic properties. See also "Dirac for his contri- 
bution. See 'electron spin resonance and 'nuclear magnetic 
resonance for one way of harnessing magnetic moments into 
useful employment. 

magnetic properties. When a substance is immersed in a 

magnetic field it has an effect which may be visualized 

(Fig. M3) in terms of the distortion of the lines of force of the 




FIG. M3. A schematic indication of the magnetic induction (magnetic 
flux density) in magnetic materials of different kinds. 

field-. In a diamagnetic material the number of lines of force is 
reduced (or the magnetic induction B is less within the body 
than in free space), and in a paramagnetic material the number 
(and induction) is increased. An alternative way of expressing 
this behaviour is to regard a diamagnetic sample as magnetized 
in opposition to the direction of the applied field, and so to 
give rise to an opposing field which partially cancels the applied 
field; conversely, a paramagnetic sample is magnetized in the 
same direction as the applied field, and the field that this 
induced moment generates augments the applied field. A 
physical manifestation of paramagnetism and diamagnetism is 
that a paramagnetic sample tends to move into a magnetic field, 
and a diamagnetic sample tends to move out of it. 

The ratio of the induced magnetic moment to the strength 
of the applied field is the magnetic susceptibility of the sample. 



136 



magnetic properties 



BOX 12: Magnetic properties 

Magnetization M = X H 

X is the magnetic susceptibility and H the field strength, 

y m vP + Y d 
A m A m A m 

paramagnetic susceptibility X p > 

diamagnetic susceptibility X^,< 0. 
Magnetic induction (flux density) 

B = HoH + fiolVl = Mo(1+xJH. 
Curve /aw 

X m = C/F; C = (ioL(i 2 m 

(u. is the magnetic moment; for example, (i 2 ~ ^V^StS + 1) 
for spin-only paramagnetism). 

Curie-Weiss law 

X m = CHT~0). 

Brillouin function 

M = NnSjiliB/kT) 

Perturbation theory expression for the molar 
susceptibilities: 



TIP: x p 



ZeVjA y , j (0llln)-(nim» j 
Xj), = -i l^ 2 -) <*" 2 > (Lartgevin-Pauli 



equation) 
See Table 20 for the Maxwell equations. 



and it is normally denoted /^ (see Box 12 and Table 20). For 
a diamagnetic material the susceptibility is negative, and for a 
paramagnetic material it is positive (in accord with the differ- 
ent directions of the induced magnetization). The susceptibility 
of all materials can be written as the sum of a paramagnetic 
susceptibility x^ and a diamagnetic susceptibility x^: 
X m = X^, + X^ (remember that x^ < and that for most 



molecules the diamagnetic term dominates the paramagnetic). 
When the molecule possesses unpaired electrons the paramag- 
netic susceptibility dominates, and it is generally found that }P 
diminishes as the temperature is raised and that in the vicinity 
of room temperature it is proportional to 1/7" (the Curie /aw). 
In a few cases the paramagnetic term dominates the diamagnetic 
even though all the electrons are paired; in such cases it is also 
found that this weak paramagnetism is independent of tempera- 
ture; for this reason it is referred to as temperature-independent 
paramagnetism {TIP} (an alternative name is high-frequency 
paramagnetism) , 

1 . Dtamagnetism . Al! molecules have a diamagnetic 
component of their magnetic susceptibility which arises by the 
applied field exerting a torque on the electrons present. The 
torque tends to drive the electrons in circles within the molecu- 
lar orbitals (Fig. M4 a), and the circulating current so produced 





FIG. Ml, (al Dtii magnetic and (b) paramagnetic currents. In la) The 
field drives a current uvithin atomic orbitals; in (b) it excites it through 
the molecular framework by mixing in excited states. 

sets up a magnetic moment and field in opposition to that 
applied. Because the diamagnetic susceptibility arises from 
processes happening within the ground state of the molecule, 
Pascal was able to draw up a table of contributions to the total 
diamagnetic susceptibility of a molecule in terms of its" struc- 
tural features which could be transferred between molecules. 
Aromatic molecules, in which there is a cyclic path for the 
electrons of the Tt-system, show anomalously large suscepti- 



magnetic properties 



137 



bilities, which are ascribed to ring currents. These ring currents 
are of particular importance in determining the form of 
"nuclear magnetic resonance spectra, but there has been some 
quarrel over their existence. 

2. Paramagnetism, The spin paramagnetism is easily under- 
stood in terms of the "magnetic moment associated with the 
electron's "spin angular momentum. In a magnetic field the 
energy of a magnetic moment depends on its orientation; 
since an electron may have one of only two orientations with 
respect to a selected axis, the application of a magnetic field 
to a collection of molecules, each with a single unpaired spin, 
lowers the energy of those with j3-spin (m = ~) and raises 
by an equal amount those with a-spin {m = + ^- Ver V 
quickly the collection of molecules relaxes into thermal 
equilibrium, and the sample then contains more £spins than 
a-spins, (Fig, M5)(the proportions are determined by the 



ttfttfftff poromoqnetc, no field 



f 4 tff4$T$4 paramagnetic, field present 



tHHtttit ,M *™<*« 



ifftfftftt K ' eal ontiferromoqnet 

FIG. MS. Magnetic materials dependent on electron spin. 

Boltzmann distribution). The spin magnetic moment lies in a 
direction opposite to the spin angular momentum (because 
the electron Is negatively charged), and so at equilibrium there 
is a net moment parallel to the applied field. The field it 
generates augments the applied field and the sample is 
paramagnetic. As the temperature is raised the Boltzmann 
distribution becomes more even between the two spin orien- 
tations because the thermal motion jostles the spin alignment, 
and consequently the net induced moment falls; a simple 
calculation applicable to room temperatures (see Question 2) 



leads to the Curie law, and a simple extension that leads to an 
expression valid at all temperatures may also be deduced {so 
long as the material does not turn ferromagnetic, see below): 
this is the Briltouin function. Both these expressions may be 
calculated for molecules with arbitrary spin. [For instance, 
molecular oxygen is a paramagnetic species with S = \, and so 
there are three orientations with different energies.) 

An important point about the role of the magnetic effects 
of the orbital angular momentum should always be remem- 
bered: in many molecules this motion is °quenched and so 
makes no contribution to the magnetic susceptibility— the 
magnetic behaviour is then referred to as spin-only 
paramagnetism. When the orbital motion is not fully 
quenched the situation is more complicated; so too is it when 
the "spin-orbtt coupling energy is large, and then the magnetic 
susceptibility cannot be calculated simply by counting spins 
and applying the Brillouin or the Curie formulas. 

Temperature-independent paramagnetism is a property of 
the "orbital angular momentum of the electrons. If there are 
low -lying excited states the magnetic field can make use of 
them to induce a migration of electrons through the molecular 
framework, and the orbital angular momentum of this motion 
gives rise to an orbital magnetic moment. This orbital motion 
differs from the diamagnetic current in so far as it arises from 
a distortion of the electronic distribution by the field, whereas 
the diamagnetic current occurs within the undistorted 
molecular orbitals. This difference leads to a current which 
percolates through the molecule in the direction opposite to 
the diamagnetic drifts within the molecular orbitals (Fig. M4), 
and so it gives rise to a magnetic moment that enhances the 
applied field. The magnetic field determines the sense of 
circulation of the current, and therefore the direction of the 
magnetization. It follows that the latter is independent of the 
jostling motion that randomizes the spin magnetic moments, 
and so this paramagnetism is temperature-independent. 

3 . Ferrom agne tism and an tiferromagne tism .Atsufficently 
low temperatures many paramagnetic materials undergo a 
transition to a state where all the spins align co-operatively 
and strongly enhance the magnetic properties of the material 
(Fig. M5). When the neighbouring spins align in the same 
direction throughout a reasonably extensive region (a domain) 



138 



magnetic properties 



of the sample, a strong magnetization is obtained and remains 
after the magnetizing field is removed. This is the ferromagnetic 
phase, and a common examples are those of iron with a transition 
temperature {known as the Curie temperature) of 1043 K, and 
of cobalt, with a Curie temperature of 1403 K (these high 
temperatures explain why 'permanent' magnets are made of 
these materials). Their magnetic susceptibility above the 
Curie temperature follows the Curie-Weiss law (Box 12), in 
which the 1/7" of the Curie law is replaced by 1/(7" +T), 7" c 
being the Curie temperature (more accurately, 1/(0^+7"), where 
8 Lis the paramagnetic intercept, which is slightly larger than 
T c : for Fe, Q Q = 1100 K and for Co, 8 Q = 1415 K). Another 
group of materials, of which NiO is a famous but far from 
unique example, shows a transition to an antiferromagnetic 
phase, in which neighbouring spins are aligned in opposition 
(Fig. M5) ; therefore the spin paramagnetism is strongly 
quenched in this co-operative state. The transition tempera- 
ture is known as the Neet temperature. The spin-spin 
interactions responsible for the alignment in both kinds of 
phase are electrostatic in origin and related to the "exchange 
energy; see Further information. 

Questions. 1. How could you recognize diamagnetism and 
paramagnetisim in a material? low is the susceptibility related to 
the induced moment? What are the dimensions of Y ? 
Consider the work necessary to insert a magnetic sample into 
a magnetic field; which is easier to insert, a paramagnetic or a 
diamagnetic sample? What is the source of paramagnetism? 
What is the difference in energy between 1 mol of electron 
spins in the ^-orientation and 1 mol in the a-orientation when 
a field of 5 kG (0'5 T) is present? What is the population 
difference when the spins are in thermal equilibrium at 300 K? 
What is the magnetization of this sample, and what is its 
susceptibility? [Try to get your units right: magnetization has 
units of A m , so does the field strength; magnetic induction, 
or flux density, has units of Wb rh" 2 , or kg s" 2 A -1 , or T; see 
Table 20.) What is the difference between the currents giving 
rise to diamagnetism and those giving rise to TIP? How can you 
justify Pascal's rules? What is the difference between 
ferromagnetism and antiferromagnetism? What are the tran- 
sition temperatures called in each case? What is the nature of 



the interaction between the spins that gives rise to the co- 
operative phenomena? 

2. Deduce the Curie law as follows. The magnetization 
(magnetic moment) of a sample in a magnetic field B is equal 
to the magnetic moment of an a-spin multiplied by the 
number of a-spins, plus the moment of a fS-spin multiplied 
by the number of 0-spins. The numbers of <xand j3 spins are 
determined by the Boltzmann distribution at a temperature 7, 
the energy of the spin with magnetic moment m being ~m^S 
as the field lies along ^. When this energy is small the 
exponential in the Boltzmann distribution may be expanded. 
Your answer should be the Curie law exhibited in Box 12. 

3. Repeat the calculation for a genera! spin S, and find the 
expression for the temperature dependence of the magnetiz- 
ation at all temperatures: this is the Brtllouin function. The 
answer is shown in the Box. Plot this function's dependence 
ong}i a BlkT. 

o 

Further information .The standard work on magnetic 
susceptibilities is by van Vleck (1932), but as it was written 
so long ago it uses rather old-fashioned language. See MQM 
Chapter 1 1 for an account of the calculation of magnetic 
properties and their interpretation in terms of currents. 
Earnshaw's book (1968) gives a discussion of many of the 
points mentioned in the preceeding paragraphs, and other 
volumes of interest are those by Oavies (1967) and Selwood 
(1956). A good introduction to co-operative phenomena is 
provided by Stanley (1971), and a tough but good intro- 
duction to co-operative magnetic phenomena is given by 
Mattis (1965). Much interest in magnetic properties arises 
from the application of "ligand-field theory to transition- 
metal ions; therefore you should look at Earnshaw and 
Harrington's The chemistry of the transition elements 
IOCS 13), Earnshaw (1968), Orgel (1960), and 
Griffith (1964). More information about magnetic properties 
will be found under "electron spin resonance, "nuclear 
magnetic resonance, and "chemical shifts. The controversy 
over the existence of ring currents may be traced by referring 
to Musher (1966). An excellent compilation of magnetic 
properties of a wide variety of materials is in §5 of Gray 
(1972), The Maxwell equations are summarized in Table 20. 



matrix 



139 



matrix. A matrix is a rectangular array of numbers. It may be 
regarded as a generalization of the concept of 'number' in the 
sense that an ordinary number is a 1 X 1 matrix and is therefore 
a special case of the general n X m matrix of numbers. In an 
n x m matrix (which we denote M} the numbers that constitute 
it (the matrix elements) may be labelled according to the row 
and column they occupy. Thus each element may be denoted 
M fC : the first subscript labels the row, the second the column. 
In a square n x n matrix the number of rows is equal to the 
number of columns, and there are n 2 elements. Square matrices 
are very important in quantum mechanics (although oblong 
matrices also occur), and our comments will be confined to 
them. 



BOX 13: Matrices 

A square matrix M is the array of n 2 elements M fc : M K is 

the element in row r, column c: 

M u "*12 ^13 ... /Wi„ 
/W 21 Af 22 M23 ■ ■ - M 2 „ 

Ms 



L M nl M r>2 M t»3 ■ ■ • M nnJ 
Addition M + N = P, where P = M„ + /V,„. 

' IC Fc fC 

Multiplication MM = P, where P fe = ^ M N . 



Examples: if M 



"♦"-ft;.*;;]'*- (::££:) 

n.b. NM=[" + * ?* + C 'l # MN in general, 
[ah *dg bh + cg_ 

Special matrices (and illustrations using the example of M 

defined above} 

L 



Diagonal matrix 

A„ - unless r = c, for example, A = 



a 




Unit matrix 

1 



all elements on diagonal = 1 
all elements off diagonal = ' 



|_o ij « 



often denoted 8 and 
re 



for example, 1 

|_U ij 

then called the Kronecker delta 
Inverse matrix M -1 : MM 1 = NT'M = 1 (see below) 
Transposed matrix 

M: M rc = M cr , for example, M = 

Complex conjugate 

M*: [M') rc = W re )\ for example, M* - fjl *.*! . 

Adjoint matrix Ivf: M* = M*; that is Mf*), c = M* cr . 

for example, M* = ,. . I . 
c J 

Unitary matrix M* = M" 1 , 

Hermitian or self -adjoint matrix M* ■ M. 

Determinant of matrix: iMl or det M; 

for example, fM I = ac — bd; 

n.b. if P = MN, P\= IMl INI. 

To find the inverse. 

1. Find IMl; if lMl# Othe inverse may be found; if IMl 
= the matrix is singular and has no inverse. 

2. Form M . 

3. Form M', where the element M' is the cofactor 

— re 

(the signed minor) of W) fC - 

4. Form the matrix M'/lMl; this is the inverse M -1 . 
For example, 1 . IM I = ac — bd, 

3. 



■'"■til]- 

^-[=4] [4 1] 



(contd.) 



140 



matrix mechanics 



= / 1 \ fac-bd -ab + abl 
\ac-bdj [cd-cd -bd + ac] 

= 1 = M" 1 M . 

To solve a set of linear simultaneous equations 
If the equations for n unknowns are 

M u x , + M t2 x 2 + ... M in x n = cy 

M 2i x 1 + Af M x 2 + , . . Af 2 „x„ = c 2 

Af,,^! + M„ 2 Jf 2 + . . . M nn x n =c n 
write them as Mx = c where x and c are the n X 1 matrices 



x = 



Xi 




■*«r 


x 2 


c = 


Cj 


>_ 




c « 



Then as M~'M = 1 and 1x = x 



x = M _1 c. 



Therefore, find M" 1 by the preceding rule, and form M"'c 
to find then unknownsx 1( ,. .x , 



Matrices may be combined together by following certain 
simple rules. If two matrices have the same dimension (number 
of rows or columns) they may be added and multiplied 
together: the rules for each type of combination are given in 
Box 13. Some matrices have special properties: these are also 
given in Box 13. Note especially that the rule of matrix 
multiplication differs from that for the multiplication of 
ordinary numbers {or c-numbers as they still are occasionally 
called): for matrices it is not generally true that the product 
MN is equal to the product NM: matrix multiplication is non- 
commutative in general. The difference MN— NM is known as 
the -commutator of M and N: the fact that the commutator 
does not necessarily disappear leads to the most significant 
differences between matrices and ordinary numbers, and, at a 
different level, is a manifestation of the differences between 
classical and quantum mechanics: see "matrix mechanics. 



1 2 
3 4 


B = 


5 6 
7 8 


C = 


3 2 + 41 

5 6 



The language of matrices has spread into ordinary 
Schrddinger quantum mechanics, for there one encounters 
integrals over wavef unctions having the form J"dT^*li^ c , 
where SI is some "operator. A convenient notation for this 
integral is £l rc , and ail the objects that may be formed from 
the wavefunctions of the system, where the labels r and c run 
over all its states, may be arranged into the matrix fi (see, 
for example, 'perturbation theory). 

Questions. Many of the techniques of matrix algebra may be 
illustrated with two-dimensional square matrices. The 
following simple problems are based on the three matrices 



A 



1. Identify the elements A u , A u, fl 21l C n , C n . 

2. Form A+B, A B, A I B+C, AB, BA, A(BC), (AB)C, AB-BA, 
AC-CA, 

3. Form A, A + , A*, A" 1 , det A, C\ C + . 

4. FormB" 1 AB,B~ 1 B, BB~ 1 . 

5. Using the rule for matrix multipHcation and identifying the 
2 X.I matrix x as[*l , express the set of simultaneous linear 
equations x \ 2y- 2 and 3x -f 4 y = 5 as a matrix equation. 
This equation will be of the form Mx=N; show that x and y 
may be found if M" 1 may be found. Find it and them. 

Further information. See MQM Chapter 6 for a summary and 
simple matrix manipulations. See Chapter 10 of Margenau and 
Murphy (1956) for a moderately complete account. Ay res 
(1962) is a good source of accounts of application of matrices 
and the way that they can be used to solve a large number of 
mathematical and physical problems. Matrices are the basis of 
the formulation of quantum mechanics known as "matrix 
mechanics and are indispensable for any thorough discussion 
of °group theory. 

matrix mechanics. The formalism of quantum mechanics 
due to Heisenberg is based on the observation that the 
position of a particle along a coordinate q and its linear 
momentum along that coordinate, p, must obey the rule 
qp-pq = ih, where ti is Planck's constant h divided by 2w. 
If one assumes that the observables of position and momentum 



molecular orbitals 



141 



obey this rule, then one obtains quantitative agreement with 
all experimental observations. Yet the rule is quite remarkable 
in content because it goes counter to all we have been brought 
up to believe in classical physics. 

In classical physics we may assign a number to the position 
(for example, a distance of 4 m from some origin) and a 
number to the momentum {for example, 2.5 kg m I* 1 ). The 
product of the numbers for the values of these observables, 
10 J s, is the same whether we calculate p X q or q X p, and so 
the difference qp-pq is zero. Heisenberg's contribution was to 
assert that the difference is not zero, but is equal to the imagin- 
ary and very small number in. It follows that q and p cannot be 
regarded as conventional numbers. Born pointed out to 
Heisenberg that the non -vanishing of the difference qp-pq 
would hold if the observables q andp were regarded as 
"matrices, for in general the product of two matrices depends 
on their order. This is the basis of matrix mechanics: instead 
of treating observables as ordinary numbers (so-called 
c-numbers, the 'c' denoting something classical) they should 
be regarded as matrices {so-called q-numbers, 'q' denoting 
something quantal) which satisfy the rule of matrix multipli- 
cation such that qp-pq = ih. When mechanical calculations are 
carried through on this basis one finds excellent agreement with 
experiment. Note that the error introduced by using the 
wrong rule qp-pq = 0, and therefore of treating q andp like 
ordinary numbers, is only of the order of h; therefore classical 
calculations are good enough when inaccuracies of the order of 
Planck's constant can be tolerated. 

Heisenberg's matrix mechanics preceeded "Schrodinger's 
wave mechanics by an insignificantly short time, and very 
quickly they were seen to be equivalent mathematical theories 
by Schrodinger himself. Today the languages of each formu- 
lation are used as convenient. The Heisenberg formulation, 
dealing as it does with the matrices, or what is equivalent, the 
"operators that represent physical observables, is often more 
convenient for formal manipulations, and the Schrodinger 
formalism (which is more easily visualizable in terms of its 
"wavefunction description of the state of a system) is often 
used for the actual calculation of the energy levels and states 
of complicated systems. 

There is a well-defined meaning to the terms Heisenberg 
picture and Schrodinger picture (or representations as they are 



often too loosely called) of quantum mechanics. The differ- 
ence between the pictures is in where the time-dependence of 
the description of a system is taken to lie. In the Heisenberg 
picture the time-dependence is borne by the operators (or 
matrices): the state remains constant but the operators that 
extract the physical information change with time. Therefore 
we observers are presented with a changing view of the system, 
and conclude that it is evolving. In the Schrodinger picture 
the operator for the desired information remains unchanging 
in time, but the wavefunction squirms around beneath it, and 
once again we are presented with a view of the system as it 
evolves in time. The difference between the pictures is simply 
one of mathematical formulation and is not of physical 
significance. There is an intermediate picture standing between 
the Heisenberg and Schrodinger viewpoints: this is the 
interaction picture, or Dirac picture . In this picture the motion 
is divided between the state function and the operator: the 
simple motion (often a harmonically varying motion) is carried 
by the operator, and the wavefunction carries the extra, com- 
plicated, but often slow, motion. This picture is very useful in 
the formulation of time-dependent "perturbation theory. 

Further information. An account of quantum mechanics 
entirely in terms of matrix mechanics has been provided in a 
short book by Green (1965). The original papers are 
Heisenberg (1925), Born and Jordan (1925), and Born, 
Hensenberg, and Jordan (1926), and English translations have 
been published by van der Waerden (1967). Bom's involvement 
is nicely illustrated in the collection of correspondence be- 
tween him and Einstein {Born 1970). For more mathematics 
see Dirac (1958), Kemble (1958), Kramers (1964), and 
von Neumann (1955). A fourth picture, to complete those of 
Schrodinger, Heisenberg, and Dirac, has been described by 
Marcus (1970). The mathematics of the first three of these 
pictures are well and simply described in §3.2 of Ziman (1969), 
§5.4 of Slichter (1963), and Roman (1965). 

molecular orbitals. The molecular orbital (MO) method 
gives a popular theoretical description of the chemical "bond, 
and is an extension of the idea of "atomic orbital to a collec- 
tion of nuclei. An electron in a molecule may be found in the 
vicinity of all the nuclei, and therefore we can regard it as 



142 



molecular orbitals 




E(H)- 



■£(H) 



— S U ^-EMO 



FIG. M6. The molecular orbital responsible for bonding in H^. 

being distributed with varying density over the nuclear frame- 
work. The °wavefunction of an electron in a molecule contains 
information about its distribution, for according to the Born 
interpretation the square of the wavef unction at any point 
is proportional to the probability of finding the electron there. 
Therefore the wavef unction for the electron in the molecule 
may be regarded as a function spreading throughout the 
nuclear framework, and its square at any point is proportional 
to the electron density. This wavef unction is the molecular 
orbital. 

The distribution of the molecular orbital should account for 
the nature of the bond: we should expect a high amplitude of 
the orbital, and therefore a high density of the electron, to 
appear where our understanding of the chemical "bond shows 
such density to be desirable, namely between the nuclei the 
electron is attempting to stick together. Think about diatomic 
compounds. In a homonuclear bond (a covalent bond between 
two identical atoms) we should expect the orbital to spread 
equally over the two nuclei; and as the bond becomes increas- 
ingly polar (in a heteronuclear bond between different atoms) 
we should expect the molecular orbital to have increasingly 
greater amplitude on one of the nuclei. In the limit of a pure 
ionic bond the molecular orbital is wholly localized on one 
nucleus. 

It is common practice to treat molecular orbitals in the same 
way as atomic orbitals are used to discuss the structure of 
atoms: the "aufbau process is applied to a set of molecular 
orbitals in order to build up the molecular electronic structure. 



As an initial example of this basic idea consider the hydrogen 
molecule. The molecular orbital responsible for the bonding is 
a symmetrical orbital extending over the two nuclei and 
having a considerable density in the region between the nuclei 
(Fig. M6). Into this orbital we insert one electron, and then 
follow it with a second; according to the °Paufi principle the 
latter must enter with its spin opposed to the first (Fig. M6), 
and no others can be accommodated. Therefore we see that 
in a very natural way the molecular-orbital theory accounts 
for the importance of spin -pairing in the formation of a 
chemical bond. This idea of the aufbau principle will be 
enlarged on when we have discussed the common approach to 
the formation and calculation of molecular orbitals. 

In principle, it is possible to imagine an extension to the 
hydrogen molecule of the calculation of the solution of the 
"Schrodinger equation for the "hydrogen atom, and even the 
direct calculation of molecular orbitals for a poly nuclear 
molecule. This is horribly difficult, and about the only place 
where it has been done is in the case of the hydrogen molecule- 
ion (H\), but even in that apparently simple one-electron case 
the calculation is not at all easy. Since chemists tend to be 
interested in molecules more complicated than Hj a scheme of 
approximation of the true molecular orbitals has been devised: 
this involves first the °Born-Oppenheimer approximation (of 
freezing the nuclei into chosen geometrical arrangements), and 
then the application of the method of "linear combination of 
atomic orbitals (LCAO). More details will be found under 
those entries; for the moment we shall simply discuss the 
application of the LCAO approach to the case of the 
hydrogen molecule and one or two other simple molecules. 

In the LCAO approximation it is supposed that the true, 
complicated molecular orbital can be expressed as a sum of 
the atomic orbitals on the constituent atoms of the molecule. 
This provides a remarkably good approximation because it 
reproduces a number of the essential features of the exact 
solution. In the case of H 2 the atomic orbitals of principal 
importance are thels-orbitals on each nucleus. The molecular 
orbital is then expressed as the sum i^i 5 + ^i 5b . This 
obviously treats the nuclei equally, and so the electron is 
spread among them equally. It also reproduces the significant 
accumulation of charge in the internuclear region. This arises 
from the wave nature of the two atomic orbitals: one Is-orbital 



molecular orbitals 



143 



FIG. M7. Bonding and antibonding orbitals in 

may be considered as a standing spherical wave centred on 
nucleus a, and the other as a standing spherical wave centred 
on nucleus b, (Fig. M7), The two waves -overlap significantly 
in the internuclear region, and if their amplitudes have the 
same sign they interfere constructively and the total amplitude 
in the internuclear region is enhanced. It follows that the 
electron density in this region is also enhanced. The energy of 
the molecule therefore is lowered by virtue of the lowering of 
the potential energy of the electrons, which, on this model, 
accumulate in the internuclear region and interact with both 
nuclei. (I do not want to complicate this description by 
invoking the role of -kinetic energy and the distortion of the 
atomic orbitals themselves. The true source of the binding 
energy must be sought in a consideration of the changes in both 
potential and kinetic energy, and the structure of the molecular 
orbitals must reflect the distortions of the atomic orbitals that 
occur when a bond is formed; see the last part of "bond.! 
Finally, it is clear that two electrons will form the bond with 
maximum stability for, according to the "Pauli principle, only 




144 



molecular orbitals 



two electrons may occupy the bonding orbital, and then they 
must have opposed spins. 

The method of linear combinations for the construction of 
a molecular orbital leads both to bonding and to 'antibonding 
orbitals: the latter are formed when the atomic orbitals overlap 
with opposite phase (sign of their amplitude), so that destruc- 
tive interference occurs and electrons are eliminated from the 
bonding region between the nuclei (Fig. M7); electrons that 
occupy these orbitals tend to drive the bond asunder. A third 
electron added to Hj would have to enter the antibonding 
orbital, and so the bond would be weakened. 

These ideas can be extended very easily to more complex 
molecules, and a diatomic molecule of considerable interest 
and importance, and to which it is instructive to apply the 
method, is oxygen ; . Our kit of parts consists of two nuclei, 
sixteen electrons, and, since this is a beginners' kit, one 
1s-orbital, one 2s-orbital, and three 2p-orbitals on each 
nucleus. The structure may be deduced as follows: 

1. The tightest-bound orbitals are the 1s-orbitals; if the two 
nuclei are pinned down at the known interatomic distance of 
0%, these two orbitals overlap to a negligible extent; therefore 
the bonding orbital they give rise to is exceedingly weakly 
bonding, and the antibonding orbital is weakly antibonding. 
This situation is illustrated in Fig. M8. 

2. The 2s-orbitals are the next tightest bound, but are much 
larger and overlap significantly. Like the orbitals in H 2 they 
form a bonding and an antibonding molecular orbital at roughly 
die energies marked in Fig. M8. 

3. Next we encounter the 2p-orbitals, and here two quite 
distinct possibilities arise: the orbitals may overlap head on, or 
broadside on. 

(a) In the former case the enhanced electron density is 
accumulated in the internuclear region, and we can expect 
a strong bond when the orbitals are in phase and a strong 
antibond when they are out of phase. We note that the 
electron distribution in such an orbital is cylindrically 
symmetrical about the internuclear axis (Fig. M8), and so 
it is termed a a-orbital, or a rj-bond (this takes its name 
by analogy with the spherically symmetrical s-orbital of 
atoms). 







Atom A Molecule Afl Atom B 

FIG. M8, Schematic energy levels and orbitals of homonuclear diatomic 
molecules of the first row elements; occupation as for O3. 

(b) The other possibility is for the broadside overlap of 
two p-orbitals. The overlap is not particularly extensive, 
and it is greatest in regions outside the internuclear axis; 
nevertheless, electrons accumulated in these regions can 
exert attractive forces on the nuclei and a moderately 
strong bond can result {and a moderately effective antibond 
if the overlap is destructive). This distribution is a jt -orbital, 
or ir-bortd (by analogy with p-orbital in atoms). 
The complete range of orbitals for molecular oxygen con- 
structed in this way is shown in Fig. M8. Into these recep- 
tacles we now inject our 1 6 electrons, and play the game 
according to the rules of "aufbau .The first tumbles down and 
down in energy until it enters 1sff. The next joins it with 
opposed spin, the next enters 1s<7*, and so it goes on, and we 
encounter no ambiguity until we have inserted 14 electrons. (A 
slight hesitation might be noticed at the filling of 2p7T, which 



molecular orbitals 



145 



takes 4 electrons; but we should remember that 2pir is really 
two molecular orbitals, one formed from 2p overlap and the 
other frorr 2p y overlap. Electron 15 we insert into the antibond- 
ing 2p7T*-arbital. Electron 16 may enter the same orbital with 
opposed spin, or it may enter the other 2p3T*-orbital of the 
pair with either the same or the opposite spin (Pauli allows 
either). What determines the outcome? The first -Hund rule 
informs us that in this situation the lower-energy arrangement 
is that with parallel spins in different orbitals; therefore we 
conclude that the "configuration of 2 is 1sa 2 Iscr* 2 2sa 2 2sfJ* 2 
2po2 2pn* 20^* 2p n*- Is there a way of testing whether 
this configuration is plausible? A powerful way is by "electronic 
spectroscopy, but another more immediate way is to note that 
the presence of the two unpaired spins leads us to predict that 
O^ is paramagnetic: see "magnetic properties. It was an early 
triumph for molecular-orbital theory that Q 2 is in fact paramag- 
netic. 

How does this structure fit in with a more elementary view 
that 2 is a molecule with a double bond, 0=0? Looking at 
Fig. M8 we can imagine that the bonding due to the 
1s-electrons ts cancelled by the antibonding nature of the 
electrons in 1 so - *; likewise the occupied 2sffand 2sa* cancel 
in effect; 2pv is occupied but 2po* empty, and so we notch up 
1 on our bonding tally; 2p x 7T and 2p tt are fully occupied, but 
2p x t! and 2p ir are both half-occupied, and so we can cut 
another notch, I", total the net bonding can hi.' ascribed to 
two net bonds, and this we can signify by 0=0, as in elemen- 



tary chemistry. It should stimulate profound respect for the 
early chemists each time their views on molecular structure- 
views formed more by introspection than by calculation— are 
confirmed by modern quantitative theory. Note too that we 
also see why a 'double bond' is less strong than two 'single 
bonds'. A single bond is generally a full a-orbital plus a full 
it -orbital, and we have seen that a TT-orbital does not have 
its extra accumulation in the prime bonding region. Earlier 
chemists ascribed this to 'strain': how right they were. A 
triple bond, which is also depicted in Fig. M9, is formed from 
one O- and two T-orbitals. The stability of double bonds to 
twisting (torsion) can also be understood in MO terms: 
rotating a CH 2 group in ethene relative to the other reduces 
the overlap between the 2pit -orbitals, the rr-bond weakens, 
and the energy of the molecule rises (Fig. M10). 

What orbitals contribute best to the formation of molecular 
orbitals? First, they must have the right symmetry: it Is no 
good attempting to form a molecular orbital from the sideways 
overlap of an s-orbital and a p-orbital, for there is no net 
"overlap. (Head-on overlap of s and p can, of course, occur.) 
That criterion satisfied we then require the orbitals to have 
about the same energy (the energy-matching criterion) and to 
be roughly the same size (to have significant net overlap). 
These criteria are explored in the Questions. The extent of 
overlap can be increased by permitting "hybridization of the 
available atomic orbitals, and the study of hydridization and 
the formation of molecular orbitals is the basis of the 



lifKjIe bond {scT Or po9 



double bond CO-4K) 




triple 
bond 
tcr+n-fn) 




itoined 
banana 
bonds 



FIG. M9. J"he formation, and two 
representations of their appearance, 
of single, double, and triple bonds. 
The bananas (which resemble the 
classical picture of strained bonds) 
are formed by taking appropriate 
sums of the 0, tr representations 
of the bonds. 



146 



molecular orbitals 




FIG. M10, Overlap and the torsional rigidity of a double bond. 

molecular-orbital approach to the discussion of molecular 
shape. 

All that has gone before suggests that molecular-orbital 
theory is a modern triumph. Is it? In a word, yes. One can level a 
number of criticisms at it, but at the expense of losing the 
conceptual simplicity of the theory these can all be overcome. 
At a basic level the molecular-orbital method underestimates 
the tendency of electrons to stay apart: the simple theory of 
H 2 , for example, ascribes too much importance to structures 
like H H~. This can be repaired by doing some -configuration 
interaction. The method is also poor at large distances: the 
separation of the hydrogen atoms in H 2 yields H* and H" in the 
products instead of just 2H; but this too can be overcome by 
permitting configuration interaction. At a different level lies 
the criticism that the molecular-orbital method does not reflect 
the chemist's view that different regions of the molecule can 
be considered for many chemical purposes as separate, isolated 
entities. The proponent of molecular-orbital theory can retort 
quickly that it is possible to take the molecular-orbitals we 
have discussed, and which spread throughout the nuclear 
framework, and manipulate them into a set of 'localized 
orbitals which can be ascribed to different regions of the 
molecule. Therefore the criticism can be circumvented by the 
application of a mathematical transformation. 

The molecular-orbital theory can be adapted to quantitative 
calculation according to the method of "self-consistent fields; 



an enormous amount of effort has been put into the calcu- 
lation of accurate molecular wavefunctions and energies by an 
extension of the methods used for atoms, and for more 
information see "Huckel method and semi-empirical methods. 
See also °ab initio. 

Questions. 1. What is meant by the term 'molecular orbital', 
and what information does such an orbital contain? What 
features should we expect a molecular orbital to possess: 
where is its maximum amplitude expected, and what can we 
say about its symmetry? Describe the changes that occur in 
the distribution of a molecular orbital as the bond it describes 
changes from pure covalent to pure ionic. What is the bonding 
energy due to over this range? What commonly used 
approximation is resorted to in order to set up a molecular 
orbital? What are the deficiencies of this method? Discuss the 
formation of the hydrogen molecule in terms of molecular- 
orbital theory. Account for the instability of the molecule 
He 2 on this basis. Set up a molecular-orbital scheme like that 
for 0^ shown in Fig. M8for all the diatomic molecules of the 
first row of the periodic table. Put the following molecules in 
order of increasing stability by referring to the diagrams you 
have just deduced: Ca, C2, Ci N^, N 2 , N 2 ; Oj, 2 , Oj,* 
^2. F?. F2." N&2, Ne 2 , Ne 2 , What is the major defect of a simple 
molecular-orbital treatment? How may it be overcome? 

2. Take the molecular orbital 1s + Is. and show that when 

a o 

it is occupied by an electron the distribution can be interpreted 
in terms of a large proportion of H + tT in the wavef unction 
(think about the square of the function). Now consider two 
electrons in Is + 15^ and two in Is — Is,., as in the structure 

a b a b 

of He 2 : what is the electron distribution in this molecule? 
Take the orbital Is + 1s. and insert one electron. Write the 

a h 

"hamiltonian for the molecule, and deduce an expression for the 
energy of the molecule in terms of integrals over the wave- 
functions. Some of these integrals can be identified with the 
integrals that occur in the description of the hydrogen atom, 
therefore the energy of the molecule can be expressed as the 
energy of the atom plus a part that can be ascribed to the 
formation of the bond. Analyse the expression in that way. 
Now insert two electrons into the orbital and so consider Hj. 
Repeat the exercise, using the appropriate hamiltonian, and 
attempt to analyse your expression. What you have probably 
forgotten to do is to allow for the *antisymmetrization of the 



molecular-orbital and valence-bond: a synopsis 



147 



electrons: you should write the wavefunction as a "Slater 
determinant, and you will discover an extra enjoyable 
contribution due to "exchange. Analyse this new result. 
3. Consider an orbital $ on atom a and 4 on atom b, with 
energies E and E respectively. Allow them to overlap and inter- 
act and make the approximation that the extent of interaction is 
proportional to the amount they overlap. Show by solving 
the "secular determinant that, of the two linear combinations 
that may be formed, one moves down in energy, and the 
other moves up; that the energy change is greater when the 
energies E a and f b are similar and when the overlap is greatest; 
that the bonding orbital is more localized on the atom with 
lower-energy atomic orbitals. These calculations illustrate the 
criteria mentioned in the text, and also show how the polarity 
of a bond reflects the relative energies of the contributing 
orbitals. 

Further information . See MQM Chapter 10 for a discussion 
of the molecular-orbital method. A simple account of 
molecular-orbital theory which fills in the details of the present 
discussion and gives many applications to chemically important 
molecules is given by Coulson in The shape and structure of 
molecules (OCS 9) and by Coulson (1961). See also Murrell, 
Kettle, and Tedder (1965), Streitweiser (1961), Salem (1966), 
Pilar (1968), Slater (1963), and Doggett (1972). Many drawings 
of molecular orbitals for numerous molecules will be found in 
Jorgensen and Salem (1973). Calculations of molecular orbitals 
will be found referred to in °self-consistent field, 'ab initio, 
and "Hiickei method. Accounts of "hybridization, "equivalent 
orbitals, 'antibonding, and 'bond extend this discussion. An 
alternative account of molecular structure is provided by the 
"valence bond theory, which should be referred to, and the two 
methods are compared under 'molecular-orbital and valence 
bond: a synopsis. 



molecular-orbital and valence-bond: a synopsis. The 

details of these techniques are given under their separate 
headings. In this synopsis an attempt is made to emphasize 
their similarities and differences. 

1. Both molecular-orbital (MO) and valence-bond (V8) 
theories seek to describe the structure of molecules, their 



shape and their energy, and the valence of the atoms that 
compose them. 

2. Both theories, at least in their simplest interpretation, 
achieve their object by leading to an accumulation of electron 
density in regions where it is most effective in interacting with 
the nuclei. This region is in the vicinity of, or actually in, the 
internuclear region. This common interpretation neglects the 
contribution to the total energy of the "kinetic energy: in both 
the MO theory and the VB theory significant contributions 
to both the potential and the kinetic energies may be ascribed 
to the distortion of the orbitals of the atoms constituting the 
molecule. This additional, but important contribution is often 
neglected in an elementary analysis of the theories: we neglect 
it here, but refer to sources in Further information. 

3. Both theories achieve the object of accumulating electron 
density in the internuclear region by recognizing that electrons 
cannot be localized on a single atom when that is part of a 
molecule. The MO theory says that if an electron can be on 
atom a with wavefunction \b (r ), which we abreviate to a (1 ), 
and can also be on atom b with wavefunction 6(1), then, 
according to the "superposition principle, its actual distribution 
must be determined from the wavefunction a(1 ) ±6 (1 ). The 
case of two electrons is obtained by dropping two electrons 
into this orbital to form ta<1) ± b (1)3 [a(2J + o(2)j . The VB 
theory approaches the problem in a different way, and says 
that if atoms a and b were well separated the state of the 
electrons (one on each atom) would be well described by the 
wavefunction a(\) 6(2), because that is the quantum-mechanical 
description of such a situation; it then pretends that the only 
difference in the function when the two atoms are at a bonding 
separation is that the electron originally on atom a may be on 
atom b, and vice versa. According to the "superposition 
principle this state of the two electrons is described by the 
function a(1) 6(2) ±6(1) a(2). Analysis of the expression for 
the energy in both the MO and VB cases leads to the con- 
clusion that the + sign in the composite functions gives the 
lower energy. The implication of this is that the electrons 
that form the bonds must enter these wavef unctions with 
paired spins (this is required by the "Pauli principle); therefore 
both the MO and VB theories account for the importance in 
chemical bonding of the electron pair. 



148 



momentum 



4. Although both methods emphasi2e the role of the 
electron pair, in practice they do so in different ways. The 
MO theory starts by ignoring the way that the electrons enter 
the molecule and calculates the molecular orbitals that may 
be formed from the available atomic orbitals. At the end of 
that work it inserts the electrons in accord with the "aufbau 
principle and, perforce, the "Paul! principle. The VB method 
concentrates on electron pairs from the outset, and calculates 
the energy of various ('canonical') structures that have all the 
electrons in the molecule paired in all possible ways. Then 
having set up all these 'perfect pairing' structures it allows 
them to interact {that is, the true wavef unction is expressed 
as a "superposition of them), and then calculates the energy 
of the best combination. This process introduces the concept 
of -resonance. 

5. Electrons are allowed to spread over the whole molecule 
automatically in the MO method, but these delocalized orbitals 
may be transformed into a collection of "localized orbitals. In 
VB theory attention is concentrated on individual bonds right 
from the beginning, and this feature is largely preserved in the 
final superimposed wavef unction. Complete derealization, of 
the sort found in 'benzene, has as its counterpart strong 
"resonance, as between equivalent Kekule' structures. 

6. Although both methods give a similar distribution of 
electrons, there are notable differences. Expansion of the 
two-electron MO given in Note 2 leads to 

a (1)« {2) + b (1) b (2) ±a (1) b (2\ ±b (Da (2). This 
differs from the VB function in the occurrence of the first 
two terms. These can be interpreted as the contribution to the 
total state of the situation in which both electrons are on the 
same atom, either on a or on b. As these extra terms appear 
with the same weight as the other terms, we conclude 
that the MO theory does not take into account the effect 
of electron correlation, the tendency of electrons to keep 
apart. There must be some probability of finding both 
electrons simultaneously on a or on b, and so we should 
expect the true function to be of the form 
8(1)6(21 + 6{1)a(2) + Xa(1)a(2) + ju6(D6 (2) 
with X, p. < 1 . This modification can be introduced by 
the method of -configuration interaction. The VB theory 
moves to the opposite extreme and forbids both electrons 



to be on a or b simultaneously; therefore it overestimates 
the role of electron correlation. It can be improved by 
adding ionic terms to the original covalent wavefunction 
(ionic-covalent "resonance). 

7. The MO method is applied quantitatively by 
feeding the electrons into approximate wavefu net ions, 
and then permitting the orbitals to distort in response 
to the electron-electron repulsions. This is taken care of 
by doing a -self -consistent field (SCF) calculation. The 
final answer is improved by permitting configuration 
interaction, MO theory has received far more attention 
than VB theory at this quantitative level because SCF 
methods are easily programmed for electronic computers; 
the difficulty of dealing with VB calculations has been 
the very large number of canonical and ionic structures 
that must be taken into account. 

Further information. See MOM Chapter 9 for more 
details of the methods and their comparison. See 
especially Coulson's The shape and structure of molecules 
IOCS 9), which is mostly MO, Pauling (I960), which is 
mostly VB, and Coulson (1961) which compares them. 
A very careful scrutiny of the nature of the chemical 
bond has been given by Ruedenberg (1962) and Feinberg, 
Ruedenberg, and Mehler (1970). See Murrell, Kettle, 
and Tedder ( 1 965) for another comparison of the 
methods. See the individual entries on "molecular orbitals 
and "valence bond for further information on each. 

momentum. In classical mechanics the momentum 
plays a fundamental role, and the same is true in 
quantum mechanics. In accord with the rules of con- 
structing quantum mechanics, the momentum, an 
observable, must be interpreted as an "operator. Once the form 
of this operator is known, other observables that depend on the 
momentum may also be expressed as operators, and so a com- 
plete scheme may be formed. The choice of the operator for 
linear momentum is of crucial importance in quantum theory, 
and one common and familiar choice for the component of 
linear momentum along the ^-axis is the differential operator 
(h/i) (9/9g). It follows from this that the linear momentum of 
a system is related to the gradient of the "wavefunction that 



multiplicity 



149 



describes its state: steep gradients correspond to high momenta. 
This aspect of the wavefunction is compatible with the 
°de Broglie relation, which states that the wavelength of a 
wavefunction diminishes as the momentum of the particle 
increases {p = A/A). When the system is described by a standing 
wave the average gradient is zero, and in such states it follows 
that the linear momentum is zero. For example, a "particle 
trapped in a one-dimensional square well is described by a 
standing wave, and its mean momentum is zero. {Classically 
that would be interpreted as multiple reflections from the walls 
of the container reversing the momentum so often that its 
mean vanished.) 

If a particle's linear momentum along an axis is sharply 
defined, its position on the axis is indeterminate: this is an 
important consequence of the 'uncertainty principle and is an 
aspect of the wave-particle -duality of matter. It can be 
understood by recognizing that a particle with definite 
momentum is described by a monochromatic wave of indefi- 
nite extent, and in such a wave, according to the Born interpre- 
tation (see "wavefunction), the position of the particle occurs 
with equal probability throughout space. Conversely, the 
formation of a "wave packet, which localizes the position of 
the particle, does so at the expense of superimposing so many 
waves of different wavelength that the momentum is broadly 
dispersed. 

Questions. 1. What aspect of the wavefunction determines the 
momentum of a state? How may the de Brogtie relation be 
justified in terms of this interpretation? Under what circum- 
stances is the momentum of a particle zero? What is the 
momentum of a particle trapped in the ground state of a 
one-dimensional square well? Why may the "kinetic 
energy be non-zero even though the linear momentum is 
zero? In what sense are the linear momentum and the 
position of a particle "complementary? 
2. By average value of the gradient is meant the "expectation 
value. Evaluate the expectation value of the linear momentum 
for particles described by running waves of the form exp ikx 
and exp(— \kx). (Note that complex-conjugate wavefunctions 
correspond to apposite momenta.) Calculate the expectation 
value for the linear momentum of a particle in a one- 
dimensional square-well potential (see "particle in a square 



well for the wavefunction). Prove from the hermiticity of 
the linear momentum operator that the expectation value of 
the momentum for a state described by a real wavefunction 
is necessarily zero. 

Further information. See MQM Chapters 3 and 4 for a detailed 
discussion of momentum in quantum mechanics. The 
fundamental role of linear momentum in quantum theory is 
described in Bohm (1951), Messiah (1961), Schiff (1968), 
Landau and Lifshitz (1958a), Dtrac (1958), von Neumann 
(1955), and Jauch (1968). 



multiplicity. The multiplicity of a "term is the number of 
"levels it possesses; that is, it is the number of different values 
of the 'quantum number 'J that may be ascribed to the term. 
When the values of the quantum numbers L and 5 of the 
term are such that L > S the multiplicity is equal to 2S + 1. 
This is because that number of values of the total angular 
momentum J may be formed by coupling the spin and 
orbital "angular momenta together {J may take the values 

L + S, L + S - 1 | L - S |). When L < S the number of 

lvalues that may be formed is 2£ 4- 1, and so under these 
circumstances the multiplicity is equal to 2i + 1. 

The numerical value of 2S + 1 is normally denoted by an 
upper left superscript on the term symbol, but it is important 
to note that this gives the true multiplicity of the level only 
when L > S. As an example a 2 D term ('doublet D') has two 
levels distinguished by J = | and J m j and written ^D^ and 
2 D 5 / 2 ; likewise a 3 P term (a triplet P term) has a multiplicity 
of three, and its levels are distinguished as 3 P 2 , 3 P[, 3 P . The 
^S term, with L = and S = \ is referred to as a doublet 
term even though it has only one level {J = \). Beware of 
sloppy usage, and always think about the relative size of 
Sand L. 

Questions. What is meant by the multiplicity of a term? How 
can it be calculated from a knowledge of the values of L and S? 
What is the significance of the left superscript on a term 
symbol? Under what circumstances does the superscript 
indicate the multiplicity? How many levels do the following 
terms possess (that is, what is their multiplicity): 



150 



multiplicity 



2 P, 3 P, *P, l S, S S, 3 D? In each case indicate the J labels 
of the levels. 

Further information . The multiplicity of a term appears 
spectroscopically as the "fine structure. For the structural 
differences of singlet and triplet terms see the entry -singlet 
and triplet states. For general aspects see -angular momentum. 
For a further discussion see MQM Chapter 8 and books on 



atomic and molecular spectroscopy: Whiffen (1972), 
King (1964), Herzberg (1944), Kuhn (1962), White (1934), 
Candler (1964), and Condon and Shortley (1963}. 
Stevenson (1965) gives a moderately simple and complete 
theoretical account of the multiplet structure of atoms and 
molecules. See Calvert and Pitts (1966) and Wayne (1970) 
for an account of the way that the multiplicity of a species 
determines its chemical behaviour. 



N 



node. A node is the place where a "wavefunction has zero 
amplitude (that is, no displacement). The node may be a point, 
a line, or a surface. For the 'particle in a one- dimensional 
square well the wavefunction has a node at the walls and at a 
number of regularly spaced points within the box, and the higher 
the energy (or higher the harmonic of the fundamental wave) 
the more nodes are present. A 1 s-orbital in 'hydrogen has no 
nodes, apart from a rather special one at infinity (see 'atomic 
orbital). A 2s-orbital has one node which should be visualized 
as a spherical surface surrounding the nucleus; as the position 
of this node depends only on the radius and is spherically 
symmetrical it is called a radial node. A 2p-orbital has no 
radial node (apart from the one at infinity), but it is divided 
into two lobes by an angular node, which is a plane running 
through the nucleus. A more complicated nodal structure 
occurs in the other "atomic orbitals, but an s-orbital always 
has no angular nodes, a p-orbital always has one, and a 
d-orbital always has two. 

The significance of the nodal structure of a wavef unction 
stems from its connexion with the "momentum or the 
'kinetic energy of the system: the more nodes in a given 
region the greater the kinetic energy. The reason for this is as 
follows. A node occurs where a wavefunction changes sign, 
and the number of sign changes in a region increases as the 
wavelength shortens. Therefore as a shorter wavelength 
implies, through the °de Broglte relation, a greater momentum, 
it follows that the more nodes present the greater the mo- 
mentum and the kinetic energy. The connexion is illustrated 
by the example already mentioned of a "particle in a square well. 

In the case of angular nodes the relevant momentum is the 



•orbital angular momentum; so we can believe that as the 
number of angular nodes increases so too does the angular 
momentum. This is confirmed by calculation, for the number 
of angular nodes is equal to the numerical value of the 
'angular momentum quantum number B, and the magnitude of 
the angular momentum is proportional to J [K(fi + 1 )] . For 
this reason the angular momentum of a d-electron (2 angular 
nodes) exceeds that of a p-electron (1 angular node). 

Questions. What is a node? What shape can it take? Can there 
be a nodal point (rather than a nodal line) for a particle in a 
two-dimensional welt? How are the nodes in an atomic 
system classified? How many nodes are there in the wave- 
function of atomic hydrogen corresponding to the principal 
quantum number n? Does the number of nodes depend on n 
and £? Does the value of m% affect your conclusions? Why is 
there a connexion between the number of angular nodes and 
the orbital angular momentum of a state? Discuss the nodal 
structure and the physical significance of the nodal structure 
of the orbitals in diatomic molecules. 

Further information. See MQM Chapter 3 for a variety of 
different systems showing nodes of various kinds. The nodes 
of functions, being places where the functions drop to zero, 
may be ascertained by locating the zeros of the function; the 
zeros of many mathematical functions are listed in Abramowitz 
and Stegun (1965). A general theorem on nodes states that the 
lowest energy level is nodeless: see §18 of Landau and Lifshitz 
(1958a). This theorem fails when many particles are present 
because Fermi holes occur in the wavefunction by virtue of 
"spin correlation. 



151 



152 



non-crossing rule 



non-crossing rule. Consider two states of an atom or mole- 
cule, and let their energy depend on some parameter P (for 
example, a bond length}. As this parameter is varied the 
energies change, and it is conceivable that a variation of P 
takes the energy of the upper energy state below that of the 
lower; that is, the energy curves cross. The non-crossing rule 
asserts that this crossing cannot occur if the states have the 
same symmetry. It follows that a variation of P leads to the 
energy variation illustrated in Fig. N1. 




FIG. N1. The non-crossing rule. Black lines correspond to states of 
different symmetry (which mav cross) and colour lines to states of the 
same symmetry (which may not). 

The rule is of considerable importance in the construction 
of correlation diagrams (see "united atom), because its 
application enables the energy of states of molecules and atoms 
to be followed as bonds are formed. 

Note that the non-crossing rule is an example of an 
-adiabatic process: if a system starts in \j/ 2 (Fig. N1 ) and the 
parameter is varied slowly, it will be in state t^ ( of the new 
system (corresponding to a different value of the parameter/ 3 
on the right of the diagram). If the motion from left to right is 
done very rapidly, the system may be in \p 2 at the final value 
of P: this corresponds to an excited state of the new system. 
Therefore we see that the non-crossing rule is applicable to 
time-dependent systems only if their motions are slow. 

Further information, A diagram like Fig. N1 may be deduced 
from perturbation theory by solving the Schrodinger equation 



for a two-state system. This is done in Chapter 7 of MQM. 
Such an analysis formed the basis of Teller's deduction of the 
rule (Teller 1937). A more abstract and earlier deduction is 
that of von Neumann and Wigner {1929). Both approaches 
have been strongly criticized, and a more rigorous proof has 
been given by Naqvi and Byers Brown (1972), and extended 
to polyatomics by Naqvi (1972). For simpler and more con- 
ventional accounts, see Couison (1961) and Herzberg (1950). 

normal modes. The number of modes of vibration of a mole- 
cule containing N atoms is 3/V-6 in general, but 3/V-5 if the 
molecule is linear. The source of these numbers is the fact that 
to specify the position of an atom requires three coordinates, 
and so the specification of the position of all the N atoms in a 
molecule requires 3/V coordinates. Changing any of the 3/V 
coordinates corresponds to changing the molecule's shape 
(bending or stretching its bonds) or to moving or rotating it as 
a whole. Of the 3/V coordinates, 3 may be chosen to be the 
position of the centre of mass of the molecule (that is, they 
specify the position of the molecule in the room), and so the 
remaining 3AV-3 must specify the position of the atoms with 
respect to the position of the centre of mass. Of these, three 
may be ascribed to the orientation of the molecule (if it is non- 
linear), and the remaining 3/V-6 must then specify the 
relative positions within the molecule of all the atoms, and 
changing them corresponds to a bending or stretching of the 
bonds, that is, to molecular vibrations. If the molecule is linear 
only two coordinates are required to specify its orientation, 
and so the number of internal coordinates is 3/V— 5. 

Consider now the case of carbon dioxide, a linear triatomic 
molecule having 9 — 5 = 4 internal degrees of freedom 
(coordinates necessary to specify the configuration of the 
molecule other than its position or orientation in the world). 
Suppose we identify one of the degrees of freedom with the 
stretching of one of the C— O bonds, and investigate the vi- 
bration of that bond. When the vibration of the bond is 
excited the other equivalent C— O bond will very quickly pick 
up its energy, because of the motion of the shared carbon 
atom, and there will be a "resonant transfer of the vibration 
from one bond to the other. The process will continue, and 
the vibrational motion transfers back and forth between the 
bonds until some external process quenches the molecular 



normalized function 



153 



villi at ion. Suppose now that instead of exciting just one of the 
bonds we were more canny and excited both equally, we 
should expect an equilibrium situation in which the vibration 
continued smoothly until it was quenched. Take, for example, 
the excitation of the symmetrical stretching mode 
O — C — K) * — >C< — 0: the carbon atom is buffeted 
equally from both sides, and there is no way (unless 
"anharmonicities are present) for the alternative combination, 
the antisymmetries! stretching mode *— 0— C-+-K3 ^ 
0-**-C— CH->-, to be excited. Nor can energy in these modes be 
transferred to the bending modes, because to do so a perpen- 
dicular force is needed (this can be provided by the 'Coriolis 
interaction arising from molecular rotation). Thus we see that 
by a judicious choice of the modes of vibration of the molecule 
we may obtain a set of independent motions: these are the 
normal modes. The four internal (vibrational) modes of C0 2 
may be chosen as four normal (independent) modes, and 
because only four internal modes exist, any vibration of the 
molecule, however complicated, may be expressed as a super- 
position of the normal modes. 

The independence of the normal modes makes their 
quantum-mechanical discussion very simple, since each of 




FIG. N2. Normal coordinates in C0 2 and H 2 (or other triatamic 
molecules). 



them may be considered to be equivalent to a single "harmonic 
oscillator of a particular mass and 'force-constant vibrating 
along some equivalent coordinate, the normal coordinate. All 
the properties of "harmonic oscillators apply in the normal 
way to each normal mode, and the "selection rule that governs 
a vibrational transition applies to the modes too. In particular, 
a mode is active in the infrared region of the spectrum (and so 
gives a line in the ordinary "vibrational spectrum) only if there 
is a change of 'dipole moment along the normal coordinate; 
that is, if the dipole moment of the molecule changes when it 
is distorted in a particular normal mode. Some of the normal 
modes of carbon dioxide and water are illustrated in Fig. N2. 

Questions. Why does a non-linear molecule possess 3N— 6 
vibrational modes and a linear molecule 3/V-5? What happens 
to the extra mode in the linear case when it is turned into a 
bent molecule? What happens when a single bond is stretched 
and then released in C0 2 , H 3 0, and CH 4 ? How is it possible to 
achieve a 'steady-state' vibration? Under what circumstances 
does the independence of the normal modes fail? What 
advantages stem from the use of normal modes? What is the 
basis of the quantum-mechanical calculation of the vibrational 
frequencies of the normal modes? What factors affect their 
frequency? If 12 C is replaced by 13 C in COj, would you expect 
all the normal-mode frequencies to change? 

Further information. See Chapter 1 of MQM For a further 
discussion of the deduction and properties of normal modes 
and vibrations. See Chapter 6 of Brand and Speakman (1960) 
for a simple discussion of the classical and quantum mechanics 
of normal vibrations. A helpful introduction to normal-mode 
analysis is given by Woodward, (1972), and more advanced 
sequels may be found in Wilson, Decius, and Cross (1955). See 
also Barrow (1962) and Ganz (1971). All these books describe 
the "group-theoretica! analysis of normal modes of vibration. 



normalized function. The Born interpretation of the -wave- 
function views \p*ir)\p{t)dT as proportional to the prob- 
ability of finding the particle in the volume element dr 
surrounding the point t; when the proportionality constant is 
unity the wavefunction is said to be normalized (or, more 
strictly, normalized to unity). 



154 



nuclear magnetic resonance: a synopsis 



Suppose the wavef unction we are presented with is ^; 
how do we proceed to normalize it? The basis of the method is 
the observation that, if the wavef unction is to be related to 
the probability distribution of the particle, then the prob- 
ability of it being somewhere in the universe must be exactly 
unity. The probability of the particle being in the universe is 
the sum of the probabilities of it being in each of the vofume 
elements dr into which the universe may be shattered; this total 
probability is the integral /dr^*^, which must equal unity. 
Therefore if in fact we find it equal to N we may normalize 
the wavef unction by dividing it by the number N y ' : the 
normalized function is N~ y ' \p and N~* is the normalization 
constant When a function is both normal and 'orthogonal it 
is orthonormal. 

Questions. What is meant by the term 'normalized function'? 
Why is it convenient to deal with normalized functions in 
quantum mechanics? What is the procedure for normalizing an 
arbitrary function? Normalize to unity the following wave- 
functions: the constant a in a universe stretching from x = 
to x = L; the functions sin*x in the same short universe; the 
function expim0 in a universe stretching around a circle; the 
function exp(— r/a ) in the whole of three-dimensional space. 

Further information, Simple examples of normalizing wave- 
functions are given in Chapter 3 of MQM. A particular 
problem arises with the normalization of functions that do 
not decay at large distances; for example, the function expixx 
oscillates for ever as x — ♦ •». Resort is then made to a special 
device (5-function normalization): this is described in §5 of 
Davydov (1965), §4 of Mandl (1957), and §1.3 of Goldberger 
and Watson (1964). 

nuclear magnetic resonance: a synopsis. The technique 
of nuclear magnetic resonance (n.m.r.) is the observation of 
the absorption of electromagnetic radiation by the magnetic 
nuclei of molecules, and in particular of protons, in the 
presence of an externally applied magnetic field. 

In a magnetic field the two orientations of the 'spin of a 
proton have different energy (a lies below j$) and transitions 
between them (inversion of the orientation, or reversal of the 
direction of spin) can be induced by an electromagnetic field 
of the appropriate 'resonance frequency. Typical spectro- 



meters employ a 15 kG magnetic field, and the resulting 
energy separation corresponds to photons of frequency 
60 MHz; recent developments have taken magnetic fields into 
the 50 kG region, where the radiation required is about 
200 MHz. These figures imply that n.m.r. is a form of radio- 
frequency spectroscopy. The experiment is generally performed 
by applying a fixed radiofrequency field to the sample, and 
varying the applied magnetic field until the radiation is 
absorbed most strongly: this is the resonance condition. A 
typical spectrum is shown in Fig. N3. 



k; 




dtemieo t shftV 



H 



PV 



FIG, N3. The structure of an n.m.r. spectrum (of acetaldehydel. In 
each case the signal is due to resonance of the starred proton. 



The information forthcoming from an n.m.r. spectrum is as 
follows: 

1 . The position of the resonance. Protons in different 
chemical environments resonate at different values of the 
applied magnetic field: this is the 'chemical shift. It arises 



nuclear statistics 



155 



because the applied field may induce local fields in the mole- 
cule, and the nuclei sense the total field. The applied field 
induces different fields in different types of chemical groups, 
and so different groups resonate with the fixed radiofrequency 
at different values of the applied field. More information will 
be found under ^chemical shift. The simplest application of 
the chemical shift is to the recognition of different types of 
groups in an unknown molecule (see Fig. N3). 

2. The fine structure of the spectrum. Under high resolution 
the lines of a spectrum are normally found to have a fine struc- 
ture. This is due to 'spin-spin coupling, in which the spins of 
the magnetic nuclei interact. As a first approximation the 
effect of the presence of the magnetic moments of the other 
nuclei is to modify the local magnetic field at the nucleus of 
interest, and the consequence of this is that it resonates at a 
value of the applied field which depends on the orientation of 
the neighbouring nuclear spins. The spin-spin coupling within 

a magnetically equivalent group of nuclei (a group with the 
same chemical shift) may be large, but it does not appear in 
the spectrum (this is a consequence of the selection rules that 
govern the spectrum). When the magnitudes of the spin -spin 
coupling and the .chemical shift are comparable the spectrum 
may take on a very complicated appearance, but when they 
are markedly different the interpretation of the spectrum is 
simple (Fig. N3). The fine structure is an excellent fingerprint 
for the identification of an unknown molecule, or for the 
determination of structure. 

3. The width of the tines; the line shape. The shape of the 
lines, and especially their width, is determined in solution by 
relaxation processes: these are described under the heading 
"electron spin resonance. The same effects operate in n.m.r., 
but as the nuclear magnetic moment is about 2000 times 
smaller than the electron spin magnetic moment its interaction 
with the environment is very much weaker and the relaxation 
times correspondingly longer (and line widths much less). 
Nevertheless, the determination of line widths and relaxation 
times is an important tool for the study of molecular motion 
in fluid solution. It is also important for the study of chemical 
motions; for example, tautomerism and proton exchange. The 
line shape is strongly affected by processes that occur on the 
n.m.r. time scale: if a motion modulates the chemical shift of 



a nucleus (for example, if a proton jumps between two 
inequivalent environments) then the line will be broadest 
when the frequency of the motion is of the order of the 
frequency difference between the two resonant positions. 

Further information. See the entries on the -chemical shift 
and °$pin-spin coupling for more information, and an idea of 
the magnitudes involved. The magnetic moments of nuclei are 
listed in Table 17. A description of n.m.r. will be found in 
McLauchlan's Magnetic resonance (OCS 1): this gives a des- 
cription of thu method and the way that ;t may be applied 
For other simple accounts see Lynden-Bell and Harris (1969), 
Jackman (1959), Roberts (1959), Carrington and McLachlan 
(1967). More details will be found in Pople, Schneider, and 
Bernstein (1959), Emsley, Feeney, and Sutcliffe (1965), 
Stichter (1963), and Abragam (1961). Recent advances are 
described in Advances in magnetic resonance, Progress in 
nuclear magnetic resonance spectroscopy, Annual review of 
n.m.r. spectroscopy, and in the Specialist periodical reports of 
the Chemical Society. For a description of rate processes in 
terms of n.m.r. see the above books, especially Chapter 12 of 
Carrington and McLachlan (1967). 

nuclear Statistics. Nuclei, like electrons and other particles, 
must satisfy the requirements of the "Paul! principle: whenever 
any two equivalent nuclei are interchanged the overall wave- 
function must change sign if they are "fermions, but not change 
sign if they are "bosons. This requirement has stringent con- 
sequences on the possible "rotational energy levels that a mole- 
cule may occupy, for the rotation of a molecule is a mode of 
motion that interchanges nuclei. Unfortunately, rotating a 
molecule also drags round the electrons, and so it is necessary 
to disentangle the exchange of the nuclei from the other effects 
that accompany rotation. We shall confine our attention to the 
hydrogen molecule, partly because it is pleasantly simple, and 
partly because it is Important through the role that nuclear 
symmetry plays in the thermal properties of hydrogen gas. 
Consider what happens when a molecule of hydrogen is 
rotated through 180°. The nuclei are interchanged, but so too 
is the orientation of the molecule as a whole, and the electrons. 
Inspection of Fig. N4 shows that it is a simple matter to 
return the electrons to their initial position in space by in- 



156 



nuclear statistics 




FIG, N4, Symmetry operations on nuclei and electrons in a 
hornonuclear diatomic. 

verting them through the centre of the molecule (no sign 
change on the electronic wavef unction for Hj in its ground 
state), and then reflecting them back across the plane in 
which the molecule was rotated (also no sign change in the 
case of the ground state of Hj ). A glance at the figure shows 
that the object of interchanging the nuclei without any net 
effect on the electrons has been achieved. 

At this point we analyse the effect of each of the con- 
stituent operators on the wavef unction of the molecule. 

The wavefunction is the product of the wavefunctions for 
the electronic state of the molecule \b „ the vibrational state 
y/ ... the rotational state \b , and the nuclear state J/ 
(of which more below). When the molecule is rotated through 
180 the rotational wavef unction changes sign if the 
"rotational quantum number,/ is odd, but does not change 
sign if it is even (think of the similar behaviour of s-, p-, d-, f-, 
. . . orbitals). Therefore from the overall rotation of the mole- 
cule, the first step of the chain, we get a factor of (—1 ) J . On 
the next two steps, the inversion and reflection of the electrons 
only, there is no change in the case of Hj (although molecules 
in states other than E may change sign), and so at stage D the 



wavefunction has changed sign by (-1) J . Finally, the spin 
orientations of the nuclei are interchanged: this takes the 
molecule from D to E, and the last stage results in the same 
molecule that would have been obtained simply by relabelling 
the two nuclei. 

When the two protons are relabelled the Pauli principle 
demands that the total wavefunction change sign; therefore 
going from A to E changes the sign of the function, and so the 
chain of operations A— B— C— D— E must also lead to a change 
of sign. So far a factor of (— 1) has been identified, but 
the step D to E may also lead to a sign change. This may be 
appreciated by considering the possible spin states of the two 
equivalent protons: the spins may be parallel or ant i parallel, 
just as in the case of two electrons. The wavefunction for the 
spins in their anti parallel configuration is a n (a)p\ (b) — 
$ n (alcc^b) (see "singlet and triplet states), and if the projections 
a n and j3 n are interchanged this function changes sign. Con- 
versely, the parallel spin arrangement (a nuclear triplet) is 
represented by the three statesa (a)a (b), a (a)p 5 (b) + 
/J lata (b), (alp^ (b), and none of these changes sign when 
the orientations are exchanged. 

Suppose that the protons of H a are antiparallel, then the 
step from D to E introduces a factor of (—1) into the circuit, 
and so the overall phase in the trip from A to E is (—1 ) + . 
But as the protons are fermions the wavefunction must change 
sign on going from A to E, and so we are forced to the con- 
clusion that when the nuclear spins are antiparallel the mole- 
cule can occupy only the rotation states corresponding to even 
values of the quantum number J. Conversely, if the proton 
spins are parallel the step from D to E leaves the sign 
unchanged, and so overall a factor of (—1 ) is obtained for the 
cycle from A to E; it follows that when the proton spins are 
parallel the molecule can occupy only the states with odd 
values of J. 

The restrictions on the quantum numbers of the occupiable 
states have two principal effects. The first is the modification of 
the appearance of the spectrum of the molecule: in a thermal 
equilibrium mixture the odd-J states are occupied three times 
as heavily as the states with even J because three orientations 
of the nuclear spins are compatible with odd values of J, and 
only one orientation with even values. This intensity altern- 
ation is characteristic of the rotational structure of the spectra 



nuclear statistics 



157 



of molecules containing equivalent nuclei. Hydrogen molecules 
with paired nuclear spins (and therefore even J values} con- 
stitute para-hydrogen, and those with parallel spins (and 
therefore odd-J values) constitute ortho-hydrogen. At thermal 
equilibrium a sample of gas contains three times as much 
ortfto-hydrogen as para-hydrogen, except at the lowest 
temperatures, where all the gas tends to occupy the lowest 
rotational level J = 0, in which case the thermal-equilibrium 
sample contains only para-hydrogen, Ortho- and para- 
hydrogen display different thermal properties at low 
temperature because of the differences in their available 
rotational energy levels. 

The attainment of thermal equilibrium can be a very slow 
process because the relative orientation of the nuclear spins 
has to be changed. A pure para-hydrogen sample will change 
only slowly at room temperature to the thermal-equilibrium 
mixture, because many of the nuclear spins must be reorien- 
tated to make them parallel to their partners. This can be 
achieved more rapidly in the presence of a catalytic surface on 
which the molecules may dissociate and then recombine with 
random partners, or in the presence of a paramagnetic molecule 
(such as oxygen or a transition-metal ion) in which the 
magnetic field of the molecule can interact more strongly with 
one proton than the other and so drive them into new relative 
orientations (see Fig, S7 on p. 218). 

The nuclear statistical effects are lost when the molecule is 
isotopically substituted, for then the nuclei are no longer 
equivalent. The molecule HD shows none of the properties of 
the kind just described. On going to D 2 a situation with 
equivalent nuclei is regained, but the deuterons are bosons and 
the intensity distribution is modified accordingly (see 
Questions). The effect of nuclear statistics on spectra and 
thermal properties may be discovered also in other molecules 
containing equivalent nuclei, such as 2 , HCCH, H 2 0, CH 3 CH 3 , 



CH 4 ; the analysis gets quite complicated but depends on the 
same arguments. 

Questions. 1 . What is the basis of the effect of relative nuclear 
orientation on the rotational energy levels of molecules? Do 
the nuclei affect the energy levels or just their population? 
Why is it necessary to consider the peculiar complicated 
scheme depicted in Fig. N4? Why are even-J states associated 
with paired nuclear spins when the nuclei are fermions, and 
odd-./ states associated with parallel nuclear spins? What 
effects does the demand of nuclear statistics have on the 
spectral and thermal properties of hydrogen? What is the 
significance of the classication of hydrogen into ortho and para 
states? What is the thermal-equilibrium concentration of the 
two species at elevated temperatures? Why is the interconirersion 
a slow process, and what can be used to accelerate it? 

2. Consider the molecule D 2 , the spin of the deuteron being 1, 
and determine which nuclear states may be associated with 
even and odd J values. What is the lowest state of the molecule? 
Show that at elevated temperatures the even:odd J state ratio 

is 2:1. 

3. Deduce that the ratio of even:odd nuclear spin states for a 
nucleus of spin / in a molecule containing two equivalent 
nuclei of spin / is (/ + 1):/. Proceed by counting the number 
of odd and even combinations of the states Im Im that may 

a b ' 

be formed. 

Further information. See MOM Chapter 10 for a further dis- 
cussion and a deduction of the general rule. A simple dis- 
cussion will be found in §4.5 of Sugden and Kenney (1965), 
and §5.13 of King (1964), who also discuss the spectral 
consequences. The thermal consequences are described by 
Davidson (1962). See Townes and Schawlow (1955) for the 
extension of these arguments to more complex molecules. 







Operators. Classical mechanics deals with observables such as 
position and momentum as functions, sometimes of each 
other, or of time, and Newton's laws of motion enable these 
functions to be discovered. Quantum mechanics recognizes 
that all the information about the system is contained in its 
"wavefunction and that, in order to extract the information 
about the value of an observable, some mathematical operation 
must be done on the function. (This is analogous to the 
necessity of doing an act, an experiment, on the system in 
order to make a measurement of its state.} Quantum mechanics 
really boils down to making the correct selection of the 
operation appropriate to the observable. 

In the simple quantum mechanics that concerns us it turns 
out that the right way to determine the momentum from a 
wavefunction is simply to differentiate it and then multiply 
the result by h/i. Thus the gradient of the wavefunction at a 
particular point determines the "momentum. The operator 
that extracts the position turns out to be simply 'multipli- 
cation by x', but this, as you can imagine, is deceptively simple. 
Once we know what the operators are for the dynamical 
variables of position and momentum we can set up the 
operators for all observables, because these can be expressed 
as functions of the two basic variables. Thus the kinetic 
energy in classical mechanics is a function of the momentum, 
namely p 2 /2m, withp 2 = p 2 + p 1 + p 2 ; therefore the 
corresponding operator can be obtained by replacing p 1 by 
(h/i} 2 (3/3x) 2 , etc.; this shows that the curvature of the wave- 
function determines the "kinetic energy. 

How does one find the operators forp and x in the first 
place? The choice is severely limited by the requirement that 



the operators be such that the values of the observables they 
yield are real numbers (the result of an observation cannot be 
a complex number}; this implies that the operators must have 
the mathematical property of hermiticity (they must be 
hermitian operators). Another requirement is that the 
operators must satisfy the rule that (xp —p x)\}/ must be 
equal to ih 4>- (We have denoted the operators corresponding to 
the observables x and p x by x andp^,) Another way of putting 
this is that the "Commutator of x andp must be ih (see 
"matrix mechanics). The latter is a very stringent requirement 
and has profound consequences; from it one may deduce the 
"uncertainty principle. 

Having found the operator for the observable of interest, 
the value of the observable for the state of the system in 
question is an "eigenstate of the operator, if the state is not an 
eigenstate the result of the experiment is determined by the 
'expectation value of the operator. 

Questions. I, Why are operators important in quantum 
mechanics? What is the operator corresponding to linear 
momentum in the x-direction, and in the /-direction? What is 
the operator corresponding to the position along the z- 
coordinate? What is the operator corresponding to kinetic 
energy, and to thez-component of angular momentum? The 
state of a system is described by the function exp \kx: what is 
the linear momentum of the state, and what is its kinetic 
energy? Another system is described by the function coskx: 
what is its linear momentum and kinetic energy? 
2. What properties must operators possess if they are to be 
satisfactory in quantum mechanics? Confirm that the 



158 



orbital angular momentum 



159 



"commutator of x and (h/i){d/dx) is ih. If we had chosen 
'multiplication byp^' to be the operator corresponding to the 
x-component of linear momentum, what would have been the 
necessary choice of operator for position? In the momentum 
representation of the operators just encountered, what would 
be the appropriate expression for the kinetic energy, and the 
Coulomb potential energy of two charges at a separation r? 

Further information. Operators are at the very heart of 
quantum theory, and so books dealing with the fundamentals 
treat operators at length. For a simple account of the basic 
theory see MQM Chapter 4. The classical account of operators 
and observabfes is provided by Dirac (1958), and more 
mathematical accounts will be found in Mackey (1963), 
von Neumann (1955), Jordan (1969), and Jauch (1968). For 
a resume, see Appendix 1 of Roman (1965). An introductory 
account of representation theory is given in Chapter 4 of 
Davydov (1965). The formulation of quantum mechanics as 
"matrix mechanics uses the properties of operators directly, 
and a succinct account is given by Green (1965). 

orbital angular momentum. The orbital angular momentum 
is the contribution to the total angular momentum that in 
classical mechanics would be ascribed to the circular motion of 
a particle around a fixed centre. In quantum mechanics it is 
found that the orbital angular momentum is quantized, and its 
values are constrained in two ways. 

1. The magnitude of the orbital angular momentum is 
confined to discrete values given by the expression 

\\-J [£(£ 4- 1)] , where E is the orbital angular momentum 
quantum number, or azimuthal quantum number, and is 
limited to positive integral values (£ = 0, 1, 2, . . . ). Thus the 
angular momentum of any body is confined to the values 
0, ft/ 2, ftV6, . . .. (Massive rotating bodies, such as a 
bicycle wheel, have angular momenta corresponding to 
£~ 10 .) In some situations (such as the "hydrogen atom) 
the maximum value of 9. is limited by the value of other 
"quantum numbers. 

2, The orientation of the direction of rotation is quantized 
(this is space quantization). The orientation of the plane of 
rotation is determined by the magnetic quantum number m% 



length VjCj4-l)' 




classical trajectory 




FIG. 01. (a) shows how a vector of length proportional to [K(.t + 1 )] li 
is related to a classical trajectory; the component on the r-axis is mh In 
(b) is illustrated the discrete orientations (with respect to the z'-axis) 
allowed by quantum theory to a particle with fi • 2. 

which can take all integral values between +£ and — fi (there 
are [2% + 1] such values). The plane of rotation is determined 
in the sense that the component of the orbital angular 
momentum about a selected axis (conventionally the^-axis) 
is limited to the value moh (Fig. 01). Therefore if the 
magnitude of the angular momentum of a particular body is 
fu/"6 (so that 9. = 2) the angular momentum about the ^-axis 
may have one of the give values — 2Ji, —16, 0, h, 2h (the different 



160 



orbital angular momentum 



signs correspond to different (classical) senses of rotation) 
{Fig. 01b). 

The component of momentum about either of the other 
axes Uf or y) is indeterminate (according to the 'uncertainty 
principle), and so if one denotes the angular momentum of a 
body by a vector I of length / [£(£ + 1 )] its projection on only 
one axis (thez-axis, by convention) may be determined at 
a particular instant: the simultaneous determination of either 
the x- or the /-component is excluded by the uncertainty 
principle. Therefore, if we still wish to represent angular 
momentum by a vector we have to draw it in a way that does 
not give the impression that we know more about its orien- 
tation than is permitted by the uncertainty principle. The best 
we can do is to draw a cone of all the possible (but indeter- 
minate) positions of the vector (Fig. 01 b), all of them of 
length proportional ton/ [£(£ + 1)] and z-components pro- 
portional to /ugh (see ^vector model). Sometimes the cone of 
possible positions is interpreted in terms of the 'precession of 
the angular momentum; but see the appropriate entry before 
you believe in such a description. 

The orbital angular momentum of a system is related to the 

number of "nodes in its wavefunction: the total number of 

angular nodes is equal to £, and the number of angular nodes 

that one encounters on encircling the z-axis is equal to Imp I. 

Thus a d-orbital has two angular nodes, and S. - 2; the d - 

' xy 

orbital has both nodes in the x^-plane, and so one encounters 
both on a circuit about the z-axis, consequently Imp I = 2. 
This connexion between angular momentum and nodal 
structure is easy to understand if one recalls the °de Broglie 
relation or, what is equivalent, recalls that the "kinetic energy 
of a particle increases as the curvature of the wavefunction 
increases. Thus a shorter- wave length wave (a more buckled 
function) has more nodes in a given length than one of longer 
wavelength. For angular momentum we are concerned with 
momentum on a circle or sphere; therefore a constant 
function (that is, a function independent of the angles and <p) 
has no nodes, is of infinite wavelength, has zero kinetic energy, 
and therefore zero (angular) momentum (or by the de Broglie 
relation has zero momentum). A function with one angular 
node (such a node lies on a diameter, and cuts a circle twice) 
corresponds to one wavelength wrapped round a circle, and 



one with two angular nodes (four nodal points on a circle) 
corresponds to two wavelengths confined to the same circum- 
ference, and so on. Therefore we see that the wavelength is 
shortened and the angular momentum is increased as we pack 
more nodes into the function. This picture of fitting waves on 
to a circle also makes clear the reason why angular momentum 
is quantized: only integral numbers of wavelengths can be 
fitted, for otherwise there would be destructive interference 
between waves on successive cycles of the ring (Fig. 02). 




FIG. 02. (a) Acceptable end (b) unacceptable waves on a ring. 

Questions. What is the classical definition of orbital angular 
momentum? How does quantization modify our view of the 
classical case? What values of the magnitude of the angular 
momentum of a rigid body are permissible? What orientations 
of a rotating body are permissible? Why is it not possible to 
represent an angular momentum by a vector in a fixed orien- 
tation? What is the connexion between the number of nodes 
in a wavefunction and its angular momentum (and which nodes 
do we count)? The number of nodes around the equator deter- 
mines Imgl: what is the significance of the difference between 
+mg and — m^? How can the quantization of both the 
magnitude and orientation of orbital angular momentum be 
explained in terms of fitting a wave to a spherical surface? For 
more questions see 'angular momentum Questions. 

Further information. See Chapter 2 of MOM for a discussion 
of the quantum mechanics of a particle on a ring and on a 
sphere: this is an introduction to the quantum theory of 



orthogonal functions 



161 



BOX 14: Orbital angular momentum 

Cfassicat definition I = r A p, 
that is, i x = VP z ~*Py 

l y =zp x -xp z 

%=*p y -yp x 

with magnitude i = (H 2 + S. 2 + S. 2 ) y '. 

X y Z 

Quantum definition 

C = r A p={h/iJrA V, 
that is. 



K = M)1y£-zfr 



2„ = (li/i) fe 



bx x bz* 

In spherical polar coordinates these become 

$ x = -lh/i) (Sin0 ^ + cotf? cos0 A) 
* 3 f\ 

£~ 2 =-Fi 2 A 2 
where A 2 is the legendrian (see Box 11). 
Commutation relations 

{%, e K l - fli^ fy, y = M x $ e , fij = ihB^ 

[£ 2 ,y =0 q = x,y,z. 
Eigenvalues and eigenfunctions 

I 2 ^J0, 0) = W + 1)h 2 ^ (9, 0) fi = 0, 1, 2 . . . 

K ^Bm < fl « *> = mh ^Sm {B - *' m » K, 8-1 HZ., 

^Em^' ^' are t ' 1e ° s P nerica l harmonics: see Table 22. 
Matrix elements. The only non-zero elements are: 



«. m + 1 1$^ l8,ro) = {W2)J [8(8 + D -m(m ±1)] 
(fi, ro ± 1 lf£ l£, m) = + (ih/2)/ [8(8 + D ~ro(ro ± 1 )] . 



Shift operators 


*! = *, 


±i8 
y 






raising operator: 








8j8,m) = rW[8(8 + 1)- 
lowering operator: 


-m(ro + l)]lB,m 


+ 1) 


JHK,m> = 


IW [£ffi + 1) - 


- m{m - 


-1)]IC,m 


-1). 



J 



angular momentum. For a more general development, see 
Chapter 6. A summary of the properties of angular momentum 
is given in Box 14, and it should be observed that the wave- 
functions for systems of given orbital angular momentum are 
the "spherical harmonics. In this connexion see Kauzmann 
0957). For a detailed account of orbital angular momentum, 
see Brink and Satchler (1968), Rose (1957), and Edmonds 
(1957). Tinkham (1964) connects it with the rotational sym- 
metry of systems. For the role of orbital angular momentum in 
chemical problems, explore -atomic orbitals, -hydrogen atom, 
and -atomic spectra and its ramifications. 

orthogonal functions, Two functions ^, and \j/ 2 are 
orthogonal if the integral _fdr»K^2 vanishes; therefore they 
are orthogonal if their -overlap integral is zero. A trivial 
example would be the orthogonality of the Is-orbital on two 
widely separated hydrogen atoms. Another example is the 
orthogonality of the 1s- and 2s- orbitals on the same atom, or the 
orthogonality of a Is-orbital on one atom to a 2pJT-orbital on 
the neighbouring atom in a diatomic molecule, 

A general result of -operator algebra is that eigenfunctions 
of an -hermitian operator are mutually orthogonal if they 
correspond to different "eigenvalues. When an eigenvalue has 
-degenerate eigenfunctions these need not be mutually 
orthogonal, but combinations that are orthogonal may be 
formed by the Schmidt orthogonalization process. In -group- 
theoretical terms, two functions are orthogonal if they belong 
to different irreducible representations of the point group of 
the system. Although orthogonality is a natural consequence 
of the type of operators one encounters in quantum mechanics, 
it is also a most desirable property because enormous numbers 
of potentially difficult integrals disappear automatically. 



162 



oscillator strength 



Questions. 1. What is meant by the term 'orthogonal function'? 
What are some examples of orthogonal functions? When can 
we be sure that a set of functions is mutually orthogonal? If 
they are not orthogonal, what process can be used to recover 
orthogonality? What group-theoretical property guarantees 
orthogonality? 

2. Consider the functions exp \m<j) in the range < 4> < 2tt: 
show that functions with different integral values of m are 
orthogonal. Show explicitly that the "hydrogen-atom 1s- 
orbital is orthogonal to the 2s- and 2p-orbitals. Show that the 
wavef unctions for a "particle in a one-dimension square well 
are mutually orthogonal. Consider the lowest-energy degenerate 
wavefunctions of a "particle in a two-dimensional square well 
and show that it is possible to find either orthogonal or non- 
orthogonal linear combinations of the two degenerate 
functions which continue to satisfy the Schrodinger equation 
with the same eigenvalue. Sketch the form of some of the 
combinations. 

3. Orthogonalize the "Slater 2s-atomic orbital to the Is-orbital 
by the Schmidt procedure. This involves forming the sum 
$u = 4>u + c\j/ tt and determining c so that the new 2s-orbital 
t/4s is orthogonal to \j/ Js , How may the Slater 3s-orbital be 
made orthogonal to </» u and ^i s ? 

Further information. See MQM Chapter 4 for the proof that 
non-degenerate eigenf unctions of hermttian operators are 
orthogonal, and for other consequences of orthogonality, For 
the group-theoretical description of orthogonality see MQM 
Chapter 5, Tinkham (1964), Bishop (1973}, and Wigner 
(1959). For the Schmidt orthogonalization process see 
McGlynn, Vanquickenborne, Kinoshita, and Carroll (1972). 

oscillator strength. The oscillator strength f is a measure of 
the strength of a transition and is the ratio of the actual 
intensity to the intensity radiated by an electron oscillating 
harmonically in three dimensions. Thus for such an ideal 
electron the oscillator strength is unity, and for strongly 
allowed transitions it is found that f lies in the neighbourhood 
of unity. Oscillator strengths for several types of transition are 
recorded in Table 8. 

The oscillator strength can be calculated from two 
directions, the theoretical and the experimental, and so its 



importance lies in the connexion it provides between theory 
and reality, as well as in its usefulness as a classification of the 
strength of transitions. 

1, The theoretical calculation of "electric dipoSe oscil- 
lator strengths is based on the expression f= Ae</nhvBle 2 , 
where B is the "Einstein coefficient of stimulated absorption. 
For electric dipoie transitions 8 is equal to cf 2 /6e fi 2 . w ' tn d 
the transit ion di pole; this implies that f= (2m /3e 2 h 2 )hvd 2 . 
Therefore, if we can calculate the "transition dipoie dfor the 
pertinent transition we can find f. 

2. The experimental determination of the oscillator 
strength is based on its relation to the "extinction coefficient 
e{v) at the wave number V, through the formula / = 4-33 X 10"' 
fdve{v). Therefore, if the extinction coefficient is known over 
the range of wavenumbers the integral provides an experi- 
mental measure of ffor the transition. 

An important theoretical rule predicts that the sum of the 
oscillator strengths for all the electrons in an A/electron mole- 
cule is equal to N: this is the Kuhn-Thomas sum rule. Thus 
overall the hydrogen atom behaves like an ideal oscillator, 
because the sum of the oscillator strengths for all transitions 
away from the ground state is unity. 

Questions. 1 . What does an oscillator strength measure? What 
is the oscillator strength of a (three-dimensional) harmonic 
oscillator? What is the sum of oscillator strengths for all 
possible transitions from the ground state of the hydrogen 
atom? State the Kuhn-Thomas sum rule for an A/-electron 
atom. How is the oscillator strength related to the extinction 
coefficient for a transition? How is the oscillator strength 
related to the transition dipoie moment? 
2. Calculate the oscillator strength for the transition from the 
rjth to the (n 4- 1)th level of an electron in a one-dimensional 
square well, and for a one-dimensional simple harmonic 
oscillator. Compute the integrated intensity of the absorption 
bands. 

Further information. For the properties of the oscil lator 
strength, its connexion with the extinction coefficient, and 
the derivation of the Kuhn-Thomas sum rule, see MQM 
Chapter 10. See also Kauzmann (1957) for a discussion and 
Eyring, Walter, and Kimball (1944). for applications in 



overlap 



163 




moderate constructive overlop shorn] constructive overlap strong destructive overlap zero net overlop 

FIG. 03. (a) Interference of waves (in black] leads to the resultant drawn in colour (since the interfering waves have different wavelengths we have 
selected only a smafl domain), [bj Shows the analogous situation for overlapping s- and p-atomic orbitals and (c) denotes the regions of overlap 
pertinent to (b|. Note that horizontal hatching implies constructive interference, and vertical hatching destructive. 



photochemistry, see Wayne (1970) and Calvert and Pitts 
(1966). 

overlap. When two waves lap into the same region of space 
they interfere and their superposition gives rise to a new wave 
with an increased amplitude in regions of constructive inter- 
ference (where the two overlapping waves have amplitudes of 
the same sign) and diminished amplitude in regions of destruc- 
tive interference (where the amplitudes have opposite signs 
and so tend to cancel) (see Fig. 03a). Since the electron 
distribution is determined by the square of the amplitude of 
the -wavef unction, the electron will be concentrated in 
regions of constructive interference and will be to some extent 
banished from the regions of destructive interference. Therefore 
the overlapping of two wavefunctions can strongly modify the 
distribution of the electron (or any other species). 

When an s-orbital is brought up to a p-orbital along the 
latter's axis {Fig, 03b) the amount of interference increases 
as the separation decreases, and if they are brought up with 
the same sign of their amplitudes the constructive interference 
increases and the electron accumulates strongly in the region 
of overlap. As the orbitals move even closer together, the region 



of overlap increases further, but the net amount of overlap 
decreases because the s-orbital begins to overlap the region 
of negative amplitude of the p-orbital on the other side of 
the node. When the two nuclei are superimposed the s-orbital 
overlaps the positive and negative lobes of the p-orbital 
equally: on one side there is constructive interference and on 
the other there is an equal amount of destructive interference. 
As the s-orbital continues on its passage through the nucleus 
the overlap gradually becomes entirely destructive because the 
amplitudes tend to cancel, and over a fairly complicated 
surface do in fact cancel to give a node. As the s-orbital moves 
away the destructive interference disappears because where 
one orbital is large the other is small, and eventually goes to 
zero. Had the s-orbita! been brought up along the line perpen- 
dicular to the axis of the p-orbital there would always have 
been an equal amount of destructive and constructive overlap 
at all distances. 

A measure of the net amount of overlap is provided by the 
overlap integral, which is zero when there is no net overlap 
(either because there is no overlapping or because there is an 
equal amount of constructive and destructive interference in 
the regions of overlap) and is unity when there is perfect 



164 



overlap 



overlapping (when the overlap of an orbital with itself is 
considered). Orbitals with zero mutual-overlap integral are said 
to be -orthogonal. The overlap integral S is calculated by 
taking the two functions at some point r, multiplying them 
together to give i/£{r)i£ (r), and then integrating this product 
overall space: S = Jarif*(r)^ b (r}. It should be clear that this 
definition of the overlap integral conforms with the properties 
we have described. 

Questions. 1. What happens when two orbitals overlap? 
Describe the change in the electron distribution that occurs 
when two Is-orbitals approach each other, pass through each 
other, and then separate {let them approach with the same 
sign of their amplitudes). Sketch the approximate value of the 
overlap integral for this process as a function of separation. 
What is the behaviour of the overlap as a Is-orbital is brought 
up to a Sd^-orbital along {a) the y-ax\s and (b) the x-axis. 
Sketch the form of the "hybrid orbitals that arise in each case 
when the nuclei coincide. 

2. The overlap of two hydrogen Is-orbitals separated by a 
distance R is given by the expression 

S(1s, Is) = (1 + R/a +RV3al) exp{-R/a ). 

a is the Bohr radius (53 pm, 0-53 A). Plot this function as a 
function of R, At what separation is S a maximum? The over- 
lap integral of a Is-orbital and a 2s-orbital or a 2p-orbita! 
approaching along the latter's axis is given by 



S{1 s, 2s) = {1/2/ 3){1 +R/a + 4flV3ag + R 3 /3al) exp(-ff/a ) 
Sits, 2po) = {/?/2a )(1 + R/a +R 2 /3a 2 ) exp{-R/a ). 

Plot these functions, and find the position of maximum overlap 
in each case. 

3, Suppose that on one of the nuclei we have a "hybrid orbital 
of the form i^2 S (r) sin£+ ^2 P (r)cos|, where sin £ and cos £ are 
mixing coefficients, and we bring up a Is-orbital along the 
axis of this hybrid. Discuss the form of the overlap integral; 
sketch its dependence on R for a mixture of 2s and 2p that 
gives the maximum overlap at all separations. Plot this 
optimum £ as a function of R . 

Further information. Overlap is of importance in all discussions 
of bonding; therefore see MQM Chapter 9, and especially 
Coulson's The shape and structure of molecules {OCS 9), 
Coulson 0961), and (vlurrell. Kettle, and Tedder (1965). The 
manner of calculating overlap integrals is described in Eyring, 
Walter, and Kimball (1944) and in more detail in McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972). Extensive 
tables of overlap integrals have been published in a number of 
places: see the list of references on p. 421 of the book by 
McGlynn etal (1972). Overlap is considerably enhanced when 
the orbitals involved are 'hybridized, and the strength of the 
bond so formed is increased. The occurrence of overlap, and 
the consequent interference effects, are aspects of the "super- 
position principle, which is one of the fundamental aspects of 
quantum mechanics. 



p 



pairing. Electrons pair when they enter an -orbital with 
opposed -spins. Electrons must pair if they are to enter the 
same orbital, for that is a requirement of the "Pauli principle. 
Electron pairs are of prime importance in the theory of the 
chemical bond, and at an elementary level it is sometimes said 
that bond formation reflects the tendency of electrons to pair. 
That shorthand must be based on the deeper remark that 
electrons will sink to lower energy if they do pair, and the 
reason for this may be found by considering the formation of 
a -bond. The first electron of a bond enters the region of 
lowest potential energy (between the two nuclei), and the 
second can enter the same region only if it has opposed spin 
and pairs with the first; because of the Pauli principle a third 
electron cannot enter the same region, and so it will be much 
less effective in bonding. The basis of this description may be 
illustrated by -molecular-orbital theory, for in order to be 
most effective in bonding an electron must enter a bonding 
orbital (which is a distribution concentrated between the two 
bound nuclei); for two electrons to enter their spins must be 
opposed; three or more electrons cannot enter this best, 
bonding orbital and are forced to enter higher-energy orbitals, 
which may be -antibonding. Hence the great importance of 
electron pairs is merely a manifestation of the fact that only 
by pairing are two electrons able to enter the lowest-energy 
orbitals and be most effective in bonding, and more than two 
cannot get into the most favourable orbital; and the tendency 
of electrons to pair is a reflection of their tendency to seek 
lowest-energy situations, 

A slightly different situation holds in atoms, for there the 
-Hund rules tell us that the outermost electrons tend not to 



pair: but here there is only one attractive centre, not two or 
more, and the important effect is the operation of -exchange 
and "spin correlation. 

Further information. See -bond and -molecular orbital for a 
guide to the role of pairing, and the -Pauli principle for its 
basic source. The -valence-bond theory is an approach to 
chemical-bonding theory that recognizes the importance of 
pairing at the outset, and develops that point of view 
quantitatively; refer to its entry for details. 

particle in a square well. A particle constrained to remain 
strictly within a particular region of space, with no seeping 
into or through the walls of the container, is a particle in a 
square well. It is so called because the confinement can be 
achieved by arranging a potential to be zero throughout the 
domain of freedom of the particle but to rise perpendicularly 
to infinity at the edges. The geometrical shape of the domain 
may take a variety of forms: a simple example is the one- 
dimensional square well, or box, where the particle can travel 
freely from to £ along the x-axis, but be nowhere else. The 
two-dimensional square well may take any shape in a plane, 
and the rectangle or the square are particular examples. The 
three-dimensional square wells include the cube and the sphere 
(which is sometimes called, without intention of paradox, a 
'spherical square well'). 

In each case the energy of the particle arises entirely from 
its -kinetic energy, because it cannot penetrate the region 
where the potential energy differs from zero. Since the 
particle is confined its energy is -quantized, and the permitted 



165 



166 



particle in a square well 



BOX 15: Particle in a square well 

Linear (one-dimensional) box of length L : 

n = 1,2 

Rectangular, three-dimensional box of sides t j, L^, Ly. 

/>i= 1, 2, . . .; n 2 = 1, 2, . . .; n 3 = \,2... .. 
Spherical box of radius R 



-^- f U\2mR 2 ) 



f l0 = 3-142, f n = 4-493, f a = 5-763, f 20 = 6-283, 
f l3 = 6-988, f tl m 7-725, . . . 

Ar as f/R;/^ is a spherical Bessel function. See §36 of 
Davydov (1965). 



energies (which are set out in Box 1 5 for various box shapes) 
are obtained from the "Schrddinger equation for the problem 
and the boundary conditions characteristic of the well. From 
the requirements that the functions be continuous and the 
probability of finding the particle anywhere outside its domain 
of freedom be zero, one deduces that the wavefunction must 
always vanish at the walls. From this it follows that although 
the permitted functions are the same as free-particle functions 
within the box, their wavelength must satisfy /?A/2 = L, with 
n an integer greater than zero; otherwise their "nodes would 
not occur at both walls. Since the wavelength determines the 
momentum by the c de Broglie relation, and since the 
momentum determines the kinetic energy, it follows that the 
permitted energy levels are confined to the values given by 
n IL . This situation is a good example of the discussion of the 
quantum-theoretical significance of "kinetic energy and the 
curvature of the wavefunction for, as we attempt to cram more 



waves into a given length, the function has to be more buckled, 
and its curvature greater. The wavefunctions themselves are 
just ordinary sine waves of decreasing wavelength, and a few 
of them are illustrated in Fig. PI. 




FIG. PI . Energy levels and wavef unctions for a particle in a one- 
dimensional square well. 



We may summarize the properties of a particle in a square 
well in the following way. 

1. In a one-dimensional box the energies are determined by 
a single quantum number n and are proportional to n 2 /L 2 (see 
Box 15)./? may take the values n = 1, 2, 3, . . ., and since 

n = is forbidden (the wavefunction with n = would vanish 
everywhere) the lowest energy is greater than zero; therefore 
there is an irremovable zero-point energy. All the states are 
non-degenerate. 

2. In a three-dimensional box the energies depend on 
three quantum numbers, and in a rectangular box of sides 



Pa sc hen-Back effect 



Li,L 2 , i. 3 are proportional to (n\ IL\) + (nllL\) + (nlli-l), 
with n ,, n 2 , n 3 allowed all integral values above zero. There 
is a zero-point energy corresponding ton l = n 2 = n i = 1. 

3. The wavefunctions are simple sine functions of 
different wavelengths {and therefore of different -kinetic 
energies). The zero-point energy corresponds to a state in 
which the particle is most probably near the centre of the 
box, and furthest from the walls. At higher quantum numbers 
the probability density is spread more evenly throughout the 
region. 

4. The energy levels move apart as the walls become more 
confining: an integral number of half wavelengths must be 
fitted into a decreasing length; therefore the wavelength 
must decrease, and so the kinetic energy increases. As the 
confining walls move apart the energy separation diminishes, 
and in the limit of infinite separation the levels form a 
continuum, and the system is unquantized. 

5. Increasing mass has an effect similar to the effect of 
increasing the size of the domain: a particle of large mass 
behaves more classically than one of low mass for a box of 
given size. 

6. A localized particle may be constructed by forming a 
suitable "superposition corresponding to a "wave packet at 
the point where the particle is located; the packet so formed 
moves in close accord with the predictions of classical 
mechanics for a particle confined to a region surrounded by 
infinitely rigid and perfectly reflecting walls. 

Questions, 1 . What is the meaning of the expression 'square 
well? What is the effect of the presence of impermeable 
walls on the allowed energy levels of the system? Why is the 
energy proportional ton 2 //. 2 ? From the classical expression 
for the kinetic energy in terms of the momentum, and the 
•de Broglie relation for the momentum in terms of the 
wavelength, deduce an expression for the energy of a particle 
in a one-dimensional square welt, and compare your answer 
with Box 15, What is the lowest energy permitted to a 
particle in a box, and what is its dependence on the size of 
the box? What interpretation in terms of the "uncertainty 
principle may be put on this zero-point energy? What is the 
mean momentum in any single energy state of the box? In 
what limit does the particle behave in a classical manner? 



167 



Calculate the lowest energy of an electron in a box of length 
1 m, 1 nm, CM nm, 1(T ls m: what is the energy of transition 
to the first excited level in each case? Why is a fly in a room a 
classical particle, to all intents and purposes (if flies have 
intents or purposes)? An electron in a conjugated polyene 
may be considered to be a particle in a one-dimensional box: 
estimate the transition energy from the/jth to the {n + 1)th 
level in a chain of N carbon atoms, 

2. The "Schrodinger equation for a one-dimensional box is 
-(1rV2m)(d 2 /dx 2 )0= E\p; solve this equation in terms of a 
function of the form Asm kx + Scos kx, and apply the 
boundary conditions that the function vanishes at x = and 
x = L, the edges of the box. Show that A may be deduced 
from the fact that the functions should be 'normalized, and 
show that states of different energy are "orthogonal. Show 
that a rectangular two-dimensional box may be solved in terms 
of 2 on|-dimensional boxes by the method of separation of 
variables (see "Schrodinger equation). 

Further information. See MOM Chapter 3 for detailed infor- 
mation about the solution of the Schrodinger equation for 
particles confined in boxes. The spherical square well is more 
difficult, but its solution is outlined in Kauzmann (1957), and, 
since it is a fair model of an electron in liquid ammonia (metal- 
ammonia solutions), see LePoutre and Sienko (1964) and 
Lagowski and Sienko (1970) for its applications and properties. 
When the barrier is not infinite the particle can seep into the 
walls, and if the potential falls outside to some finite value the 
particle might 'tunnel through the barrier: situations of this 
kind are discussed and solved in §32 of Davydov (1965), 
Schiff (1968), Messiah (1961), and Landau and Lifshitz 
(1958 a). Gol'dman and Kryvchenkov (1961) work through a 
number of problems involving barriers. See also Chapter 1 1 of 
Bohm (1951 ) for a good discussion of square potentials. 

Paschen-Back effect. The Paschen-Back effect is the de- 
coupling of spin and orbital angular momenta by an applied 
magnetic field. 

Consider an atom with both "spin and -orbital angular 
momenta, s and I; these are coupled together by the "spin- 
orbit coupling interaction and form a resultant angular 
momentum which is represented by the "vector j. Both s 



168 




FIG. P2. Paschen-Back eCfed. 

and I precess around their resultant, and in a magnetic 
field of moderate strength the resultant vector also 
precesses about the field direction (Fig. P2 a): the energy 
of this interaction gives rise to the ""Zeeman effect. As the 
strength of the applied field is increased the strength of its 
interaction with the spin and orbital "magnetic moments 
becomes so great that it begins to overcome the spin-orbit 
coupling. At this point the spin and orbital moments begin 
individually to precess around the direction of the field, 
and the spin-orbit coupling is broken; the motion is complex 
because the spin-orbit coupling and the applied field are in 
competition for the two moments. If the field is made 
sufficiently strong (of the order of tens of kilogauss) the 
battle is resolved in its favour, and the state is one of almost 
pure precession of each type of momentum about the 
field's direction (Fig. P2 b). The Paschen-Back effect has 
succeeded in decoupling the momenta. 

The spectral consequence of the effect is that the elec- 
tronic transitions occur in a simple fashion: the optical 
field interacts with the orbital angular momentum and 
causes transitions which are independent of the spin 
direction. The anomalous 'Zeeman effect, which depends 
on the interplay of spin and orbital moment effects, is 
replaced by the normal Zeeman effect, characteristic of 
systems without spin. 

The term Paschen-Back effect is also used for other 
situations in which the spin and orbital momenta are 



decoupled by a field; one example is the effect of the 
axial nuclear Coulombtc electrostatic potential on the 
momenta in a diatomic molecule; its effect is to modify the 
"Hund coupling case. 

Questions. What is the Paschen-Back effect? Discuss the 
effect in terms of the "vector model of the atom. Why does 
the effect simplify the anomalous Zeeman effect and 
replace it by the normal effect? If the spin-orbit coupling 
energy in an atom is of the order of 100 cm" 1 estimate the 
strength of the applied field required to decouple the 
angular momenta. 

Further information. See the discussion of the 'Zeeman 
effect and MQM Chapter 8. See also §11.3 of Herzberg 
(1944) and §111 F3 of Kuhn (1962) for detailed accounts 
of its spectroscopic consequences. 

Paul! principle. The Pauli exclusion principle states that no 
more than one electron may occupy a particular state: a con- 
sequence of this is the "aufbau principle which underlies the 
periodicity of the elements; for an atomic orbital labelled 
n, 6, mn, may be populated by no more than two electrons and 
these must differ in the quantum number m (and so must have 
opposite spin orientations). 

The exclusion principle is a special case of the full Pauli 
principle which makes a general statement about all particles. 
The Paul! principle starts from the view that if two particles 
are indistinguishable, when they are interchanged the calcu- 
lated properties of the system must remain unchanged. In 
particular, since the particle density is proportional to i^l 2 
(see -wavef unction), when two indistinguishable particles are 
interchanged (the first put where the second was and vice 
versa) the particle density, and hence l^l 2 , must not change. 
This implies that \jj itself can either change sign when the 
particles are interchanged, or not change sign; no more 
complicated change is permissible. The change of sign, or lack 
of it, must occur for all possible pairs of indistinguishable 
particles in the system when the two particles of each pair are 
interchanged. A fundamental distinction which determines 
whether or not the wavefunction changes sign is between 
•fermions (which are particles with half-integral spin, such as 
electrons, protons, neutrons, 13 C nuclei, and 3 He) and "bosons 



(which have integral spin, such as 2 H, 4 He, 12 C, and photons). 
The principle states that if a collection of identical particles 

1,2 n is described by the wavef unction ip{r h r 2 r ) 

then the form of this function must satisfy a stringent 
requirement: If the particles are fermions the function must 
change sign whenever the positions of any two particles are 
interchanged; whereas if the particles are bosons the function 
must not change sign. Another way of saying this is that under 
particle interchange the total wavef unction for identical 
fermions must be antisymmetric, and for identical bosons it 
must be symmetric. 

The principle is adequately illustrated by considering a two- 
particle state Mr,, t 2 ), or ^(1, 2} for short: if the particles are 
fermions Nature demands that i£{2, 1) = -^1, 2). The impli- 
cation of this for electrons (fermions) is as follows. Suppose 
we have a state \j/ a which can be occupied by electron 1 and a 
state ^ which can be occupied by electron 2, then the total 
system is described by a function of the form $ (rj^ (r 2 ), or 
^,(1)^(2) for brevity. But this function does not satisfy the 
Pauli principle because it is neither symmetric nor anti- 
symmetric in the labels 1, 2 (it is unsymmetric). We can turn 
it into an acceptable function for electrons by replacing 
^,{1)^(2} by \pJ1)\p b i2) - lA a (2)^ b (T), for when 1 and 2 
are interchanged the sign of the function changes. This 
function demonstrates why only one electron can exist in a 
given state: if the states ^ a and ^ b were identical, so that 
both electron 1 and electron 2 were in the same state, the 
form of the function that satisfies the Pauli principle would 
be ^0)^(2} -^(2)^(1), which vanishes: this shows that 
multiply-occupied states vanish and that the exclusion 
principle is a special case of the full principle. If instead of 
electrons we were dealing with bosons, the appropriate 
two particle state would be ^ a (1)^ b (2) + ^ (2)^ (1), and 
this does not vanish if $> = $ < therefore bosons do not 
satisfy the exclusion principle because multiply-occupied 
states are permitted (a large number of photons may occupy 
the same state and give rise to an intense monochromatic 
light beam). Remember that the Pauli principle applies to the 
total state of the system, so that ij/ g and & represent space 
(orbital) and spin states. 

The Paul! principle was introduced in order to account for 
the spectrum of helium, for an analysis of its spectrum showed 



penetration and shielding 



169 



that many expected "terms were absent, and their absence 
could be explained on the basis that two electrons on a given 
atom could not occupy the same state. The principle can be 
given a theoretical foundation, and Pauli, by considering the 
problem relativistically, and demanding that the energy of a 
system be positive, showed that all particles of ha If- integral 
spin must have antisymmetric wavefunctions and behave as 
we have described. Possible exceptions to the principle are 
quarks, which are peculiar particles used in one theory of 
elementary particles; but as they have not yet been observed 
experimentally they may be figments of the imagination, and 
particles, not figments, are required to satisfy the Pauli 
principle. 

Without the Pauli principle matter would not be rigid; 
likewise, if electrons had no spin (and were bosons) matter 
would have less bulk, everything would be denser, nothing 
rigid, and everything very sticky. 

Further information. The experimental basis of the Pauli 
principle is described in MQM Chapter 8. A product of 
functions may be made to accord with the principle by 
writing it as a "Slater determinant, and the discussion is taken 
further in that entry and in the one on "antisymmetric 
functions. Problems too will be found there and in Further 
information. Pauli deducing his principle may be observed in 
Pauli (1940); for quarks see "fermions. 

penetration and shielding. Like Castor and Pollux, 
penetration and shielding seem inseparable twins: so they are 
in application but not in contemplation. Let us fix our ideas 
by considering the sodium ion Na + , which consists of a 
strongly charged nucleus (Z= 11) surrounded by ten electrons 
filling the K and L shells (the -configuration is 1s 2 2s 2 2p 6 }. 
Drop an electron into the 3s- "orbital and observe that, since 
the s-orbitals all have a non-vanishing probability of being at 
the nucleus, the electron penetrates the surrounding electrons 
to a small extent, and with a small but non-zero probability 
may be found in the vicinity of the nucleus. In that region it 
will have a low potential energy and be stabilized. If instead of 
dropping it into a 3s-orbital we contrive to deposit it in a 
3p-orbital, the electron is unable to penetrate so closely to the 
nucleus {p-orbitals have modes at their nucleus), and so it does 



170 



perturbation 



Election 
density 




Distance from nucleus 



FIG. P3, The coloured lines show the 3s and 3p radial-distribution 
functions superimposed on K, L shell electron density. 

not attain the region of lowest potential energy (Fig. P3). That, 
then, is penetration; we can believe that s-orbitals might lie 
lower in energy than p-, d-orbitals on account of their closer 
approach to the nucleus. 

Close approach is not of itself sufficient to lower the energy, 
because in the -hydrogen atom all the orbitals with the same 
principal -quantum number carry the same energy, irrespective 
of whether they are s, p, or d. In order for penetration to have 
an effect it is necessary to eliminate the peculiar property of 
the pure Coulomb interaction which is responsible for the 
unique properties of the hydrogen atom. This is achieved by 
the shielding effect of the other electrons that are present in 
the case of many-electron atoms (see Fig. P3). In these the 
potential experienced by our extra electron is that of a nucleus 
of charge Ze(Z= 1 1) at small radii, but at larger radii, when 
the test electron is outside the core, the potential is more 
characteristic of a nucleus of charge e, because the (Z — 1 ) 
electrons have shielded all but one unit of positive charge. 
Therefore the potential of the shielded nucleus drops off far 
more rapidly than the Coulombic law would entail, and so the 
peculiar characteristics of the Coulombic potential are 
eliminated. Consequently the s-, p-„ and d-electrons possess 
different energies. 

It should be clear that the different orbitals have different 
energies on account of the shielding effects, and the order of 



their energies and their separation depends on the extent of 
their penetration of the core. 

Questions. Which orbitals penetrate most closely to the 
nucleus? What is the essential role of shielding? What 
accounts for the fact that s-electrons generally lie lower in 
energy than p-electrons, and p-electrons lie lower than 
d-electrons? What is the order of 'ionization potentials for 
s-, p-, and d-electrons? 

Further information. See MQM Chapter 8 for a description of 
the extent and role of penetration and shielding. Simple 
diagrams are given there, and in White (1934) and Herzberg 
(1944), of the extent of penetration of various Inner shells by 
various outer electrons. For a discussion of the screening 
constant see "Slater atomic orbitals. The effects of pen- 
etration and shielding are most important for the structure of 
the periodic table, for they influence the properties of the 
elements through the 'aufbau principle and the -ionization 
potential. These matters are discussed by Puddephatt in 
77je periodic table of tfie elements (OCS 3) and by Phillips 
and Williams (1965). 

perturbation. Most systems of interest are described by 
-Schrodinger equations too difficult to solve exactly; but 
inspection of the problem often reveals that a simpler, 
solvable system closely resembles the true, intractable system. 
If the "wavef unctions and energies of this simpler system are 
known it is possible to adjust them so that they are distorted 
in the direction of the true wavefunctions and energies. If the 
true system resembles the simpler system very closely the 
amount of distortion required is very small, and is a mere 
perturbation of their form. The modification of the simple 
wavefunction can be achieved by mixing into it other wave- 
functions of the simple system in the appropriate proportions, 
and perturbation theory provides the recipe for the mixture. 
At the same time, perturbation theory shows how to cal- 
culate the additional terms that must be added to the energy 
of the simple system to yield the energy of the true system. 

First-order perturbation theory teaches that the wave- 
function of the simple system \p should be modified to 
&o + C|& + c 2 4/ 2 + - - n w here tne $„ are tne v a rious waue " 
functions of the model system corresponding to the energies 



£,, and where the coefficients^ are determined by the 
ratio of the strength of the perturbation (which is essentially 
the energy difference between the true and the simple systems, 
and is actually the 'matrix element of the perturbation 
hamiltonian-the difference between the true -hamiltonian 
and the simple hamiltonian) to the energy separation E — E 
The wavefunction obtained in this way is the first-order 
wavefunction. 

An example to fix our ideas at this stage could be the 
distortion of the ground state of the hydrogen atom by an 
applied electric field (the "Stark effect): the simple system 
would be the hydrogen atom in the absence of applied fields 
(the wavefunctions and energies, of course, are known), and 
the distortion ("polarization) of the atom by the applied field 
could be taken into account by adding some 2p -orbital into 
the ground state, and then some 3p^ -orbital, and so on. The 
coefficient of each orbital is determined by the ratio of the 
perturbation energy (in this case the electrical interaction 
energy ezE, which comes from the expression -d.E for a 
dipole d in our electric field E) to the energy separation for 
each orbital (£ ls -£ 2p , E ls -f , p , . . .) , The proportion of 
the orbital in the mixture is determined by the square of the 
mixing coefficient (see "superposition principle), and so it is 
clear that only those orbitals lying fairly close in energy to the 
1s-orbital will be mixed significantly. This is a general result of 
perturbation theory: the greater the energy separation, the fess 
the mixing. It follows that if all the energy levels lie far above 
the ground state, the simple system cannot be distorted very 
much, and so it will resemble very closely the true system. 
Conversely, if the perturbation is too strong (the true and 
simple systems being very dissimilar) a large amount of mixing 
may occur and the first-order theory might be inappropriate: 
the solution is generally to choose a better model system. 

It should be noticed that the form of the true wavefunctions 
emerges automatically from the perturbation-theory 
machinery, for the recipe generates the correct distortion. This 
is in contrast to the "variation approach to the approximation 
of wavefunctions, where the final form depends on the 
original guess of the form of a sufficiently flexible function. 

The modification of the energy due to the perturbation 
may also be calculated, and although the first-order energy is 
sometimes sufficient, it is normally necessary to calculate the 



perturbation 171 

second-order energy. The former is calculated by taking the 
perturbation energy and calculating its average value over the 
undistortsd wavefunctions of the simple system. The second- 
order energy correction takes into account the distortion of 
the simple wavefunctions by the perturbation: first the wave- 
function is distorted by the application of the perturbation, 
and then the average value of the perturbation energy is 
calculated over this distorted wavefunction. The name 
'second-order' indicates that the perturbation is involved 
twice, 

A helpful analogy, which enables one to appreciate how 
perturbations operate, is the mutilation of a violin string by 
suspending from it a number of small weights. The weights 
hanging from the nodes affect neither its motion nor its 
energy, but those hanging from the antinodes (the points of 
maximum displacement) may have a profound effect on its 
vibrational energy and waveform. The first-order energy 
correction is found by averaging the effect of the weights (the 
perturbation) over the wavefunction (in this case the dis- 
placement) of the system. The weights also distort the wave- 
form of the string: the nodes are slightly shifted and the pure 
sinusoidal shape lost. This distorted mode can be reproduced 
by taking a suitable superposition of the harmonics of the 
unladen string, and if the weights are not ton big it should be 
clear that only a few of the harmonics need be incorporated 
to give a good representation of the distortion. The second- 
order energy correction to the vibrational energy of the string 
can be found by averaging the effect of the weights over the 
distorted waveform. This second-order correction should be 
added to the first-order correction, and their sum added to the 
original vibrational energy of the unladen string, in order to 
find a good approximation to the true energy of the 
modified string. 

The order of magnitude of the correction to the energy 
brought about by a perturbation of energy P is ~/> for the 
first-order correction, and ~P 2 /A for the second-order 
correction, A being a typical energy separation of the undis- 
torted system. Often the first-order correction disappears 
identically (on grounds of symmetry). The first-order 
correction to the wavefunction yields coefficients of order 
P/A, and so the proportion of states admixed is of the order 
P /A 2 . The application of perturbation theory is normally 



172 



perturbation 



valid so long as P is much smaller than A: that is, so long as 
\P/A\-% 1. It is important to note that the second-order energy 
can be obtained from the first-order wavef unction (in general 
the nth-order energy can be obtained with knowledge of the 
(n — 2)th-order functiont); thus jubilation at obtaining a good 
energy should be tempered with the reflection that the wave- 
function itself might still be very poor, Alternatively, the more 
sanguine will reflect with some satisfaction that such a good 
energy can be obtained from such a poor wavef unction. (For 
example, a wavefunction correct to within 10 per cent can give 
an energy correct to within' 1 per cent.) 

The algebraic recipes for perturbation theory are set out in 
Box 16. 

Time-dependent perturbation theory allows the perturbation 
to vary with time and permits the calculation, as an important 
application, of the effect of the perturbation caused by a light 
wave. The distortions change with time and the admixture of 
the excited states of the system may be interpreted as tran- 
sitions from the ground state to another state. See -transition 
probabilities. 

Questions, 1. What is a 'perturbation'? What does it affect? 
How may the effect of the perturbation of the wavefunction 
be taken into account? What is the order of magnitude of the 
coefficients of the admixed functions, and in what proportion 
are they mixed? How is the first-order correction to the energy 
calculated? How is the second-order correction calculated? 
Why do they differ? What action should you take if you find 
that the wavefunction of the simple system has to be severely 
modified? What is the order of magnitude of the first- and 
second-order energy corrections and the first-order wave- 
function distortion when a perturbation of energy equivalent 
to 10 J mot' 1 is applied to a system in which the energy 
separations are of the order of 1 kJ mof l ? A mass of 1 kg 
hangs from a spring of force-constant 1 N m" 1 : how will its 
motion be modified by the addition of a 100 g mass 
ls=9-8m s~ 2 \? Now consider a similar quantum-mechanical 
problem: let a 2,0 Po atom be oscillating against an effective 
"force-constant of 40 N m" 1 , and let it be in the vibrational 

t Really, if the lower orders don't set us a good example, what on earth 
is the use of them? (Oscar Wilde. The importance of being earnest, 
Actl.) 



BOX 16: Perturbation theory 

We suppose that we know the eigenvalues and eigenfunctions 
of the hamiltonianW (0) : 

wloi^W = £(°)iM > 

r n n r r» ' 

and we require the energies and wavefunctions of a 
hamiltonian W: 

where W< " is a first-order correction (of order X in some 
small parameter X) and rV' 2 ' is a second-order correction 
(of order X 2 ). The true energies and wavefunctions are 
written 

f = £<0 l + f (l) +f {2) + _ 

m m ttt m 

\b =li/< »-|- 0< l >+ lir< 2 >+.:„ 

*vn r m 'm r m 
Zeroth-order energy and wavefunction 

E m =<^\H l0) l^ 0) > corresponding to ^ 0) . 

First-order correction 

f i !) = <C lwl,,| C ,) 



m 






0**> = T,' e ^"l, where c = J ll*St 

[ m n ) 

Second-order correction 

+ *\ <c-4°>> ffl r 

Time -dependen t perturba tion theory 

We suppose that H = H w + H ll Ht) 

and write 

* m W = E c „ W 1 W* *1 0) W = tfT exp (-i£<°>f/r.). 

(if *^m^ it n n ft ft 

n 

Then 
0„{t) = e n i0)-(mZfl dtWW&W exp fa^ . 



phonons 



173 



where 
and 



w^ , ( f ')=<c w(i)|f ' )i ^r ) 

nk n k 



For weak perturbations applied to a state in which 
initially c.(0) = 1, c f {Q) * (only state / occupied), then to 
first order: 

c ; (t) = 1 

e,w = -(r/h>£df'wk»(r')wcp i^r'. 

"1 — ex p \\{iti fl + to)t] 1 — exp [ i (uy f; — cj) f f| 



c f M=V fi 



ll(CJ ff + Oj) 



H<*> fj ~ CO) 



and if ui f . - oj < GJ ff + to the probability of being in state 
fis 



4V V 



sin 2 |(oj ff - 6j}f, 



ground state. Let it emit an a-particle. What is the order of 
magnitude of the probability of finding the resulting ^Pb 
atom oscillating in its ground state? 

2. The quantum-mechanical expressions for the perturbation 
corrections to the energy and wavef unction are given in 
Box 16. Take a "particle in a one-dimensional square-well 
wavefunction and add to the system a perturbation of the 
form —qx, <x < L: find the first-order energies and wave- 
functions. Using the same model, select a system that has a 
flat potential within the walls except for a small rectangular 
dip of depth D and width W. Set the centre of this dip at the 
centre of the well, and let W< L. Calculate the first-order 
energy correction to a particle in the ground state [n = 1 ), 
and then in the first excited state {n = 2). What do you notice 
about the extent of correction in each case? Is the difference 
also reflected in the correction to the wavefunction? Now 
slide the centre of the dip to* = LIZ; what happens? 
3. Estimate the extent of distortion and the correction to the 
energy that results when an electric field of 10 6 V m~' is 
applied to a ground- state "hydrogen atom. 

Further information. See MQM Chapter 7 for the details of 
perturbation theory. Perturbation theory is one of the most 



important methods for calculating atomic and molecular 
properties: therefore see MQM Chapter 11, Eyring, Walter, 
and Kimball (1944), Hameka (1965), Davies (1967), and 
Kauzmann (1957). The mathematics of perturbation theory 
and its recent developments are described by Hirschfelder, 
Byers-Brown, and Epstein (1964) and by Wilcox (1966). The 
last two references describe the differences between the 
Rayleigh-Schrodinger perturbation theory (which is the scheme 
set out in the Box), and the Wigner-Briiiouin perturbation 
theory. The convergence of a perturbation expansion to the 
exact energy is a frisky problem, slightly tamed by the 
Rellich-Kato theorem described on p. 6 of Wilcox (1966). 
Time-dependent perturbation theory enables, among other 
things, the evolution of a wavefunction to be calculated as a 
function of time; see -transition probability, Chapter 7 of 
MQM, Davydov (1965), and Heitler (1954). An account of 
Heftier 's theory has been given by Hameka (1965). A recent 
review of time-dependent perturbation theory is that of 
Langhoff, Epstein, and Karplus (1972). 

phonons. Just as -photons are "quantized vibrations of the 
electromagnetic field so phonons are quantized vibrations of a 
crystalline lattice. Imagine first a linear chain of atoms which is 
vibrating in a low-frequency mode (Fig. P4): this vibration is 
quantized and may possess only discrete amounts of energy; 
if its frequency is V its energy must be some integral multiple 
of hv. Instead of exciting a single vibrational mode through its 
successive evenly-spaced quantum levels it is possible to regard 
the rising energy of the system as resulting from the addition 
of hypothetical particles to that state; if n particles enter a 
state of characteristic frequency V the energy of the system 
rises by nhv. This particle picture of excitation of vibrations 
is the basis of the concept of a phonon: a phonon is a 
quantum of excitation of a specified frequency, and may be 
envisaged as a particle of energy At 1 added to the system. 

In a complex solid, phonons of different frequency exist 
(just as in the electromagnetic field light of different fre- 
quencies exists). As in the case of light the phonons in a solid 
may be polarized, but as well as transverse phonons, in which 
the lattice atoms are displaced perpendicular to the propagation 
direction (Fig. P4 a), longitudinal phonons, in which the dis- 
placement is along the direction of propagation, as in a sound 



174 phosphorescence 

framvene mode 




longitudinal mode 



wave in a fluid, rnav also exist (Fig. P4 b). A further compli- 
cation may be illustrated by the case of an tonic lattice con- 
taining positive and negative ions. First imagine the vibrations 
of the two interpenetrating lattices as independent: now con- 
sider the combined system. Two situations may arise (see 
Fig, P5). In the first the lattice of positive tons moves in phase 
with the lattice of negative tons: this gives rise to the 
acoustical branch of the phonon spectrum. In the second the 
phase of the displacement of the two lattices is opposite: this 
gives rise to the optica/ branch. The latter name arises from the 
modulation of the lattice dipole moment during the out-of- 
phase movements of the two sublattices, and the fact that a 
light beam can interact with this oscillating dipole and so 
stimulate that particular branch of the lattice vibrations. 



ccowric branch 
Q © 




n- 



©r © e © 

© © 

€X>€MD 





-® 
© © 

optical branch 




® 



-® 



-® 



-© 



FIG. P5. Acoustic and optica) branches of the same wavelength. Note 
that the dipote changes only in the latter. 



3+i 
2+i 
l+l 
0+t 



Energy/hy 



3 phonon 5 



2 phonom 



I phonon 



no phonons preienr 



FIG. P4. Transversa and longi- 
tudinal modes □( the same wave- 
length: the extent of excitation 
can be expressed as the number 
of phonons present. 



Further information, A good introduction to lattice 
vibrations is given in Kittel (1971) and a more advanced one 
by Kittel (1963) and Ziman (1960, 1972). The role of 
phonons in determining the electrical, optical, and thermal 
properties of solids is discussed in these books. For a simple 
account of the matter see Jennings and Morris's Atoms in 
contact (OPS 5). 



phosphorescence. When a phosphorescent material is 
illuminated it emits light, and the emission may persist for an 
appreciable time even after the stimulating illumination has 
been removed; in this sense phosphorescence differs from 
■fluorescence, for in the latter the emission ceases virtually 
instantaneously. There is also a mechanistic distinction 
between the two processes, and this is connected with the 
persistence of the emission. 

Phosphorescence occurs by the following mechanism: the 
ground-state molecule, in which all the spins are paired (and 
which is therefore a 'singlet state Sn), absorbs the incident 
light and makes a transition to an upper singlet level. The 
electronic excitation is accompanied by a vibrational 
excitation (see Fig. P6), and this vibrational energy is trans- 
ferred to the surrounding molecules by the type of processes 
described in "fluorescence. Indeed, the molecule is well along 
the path that leads to fluorescence; but if the vibrational 
deactivation is not too fast another process may intervene. 
Let there be a triplet state of the excited molecule (in which 
two spins are unpaired): this is illustrated as the curve T, in 
the diagram. There is a non-vanishing probability that the 
molecule will switch from the singlet state to the triplet as 



phosphorescence 



175 




FIG. P6, The mechanism of phosphorescence, 

it steps down the vibrational energy ladder (this is the 
inter-system crossing, ISC). At the foot of the ladder it finds 
itself trapped. It cannot radiate its electronic energy and drop 
to the ground state because that involves a singlet-triplet 
transition (which is forbidden by the "selection rules). It 
cannot clamber back to the crossing point and then step down 
the singlet ladder, because the collisions with the lattice cannot 
supply enough energy. It cannot give up its electronic energy 
to the surrounding molecules by a radiationless transition, 
because we have supposed that even the vibrational deacti- 
vation is weak, and that involved a smaller energy {which is 
easier to remove). If the foregoing description were true, the 
molecule would be stuck in the upper triplet. But the remarks 
are not strictly true, like most remarks, and the important 
fallacy is that the -singlet-triplet transition is forbidden. If it 
were strictly forbidden the molecule would have been unable 
to cross from the singlet state to the triplet. The fact that it 
did cross implies that there is enough "spin -orbit coupling 
present to break down the singlet-triplet selection rule, and so 
this becomes weakly allowed. But as it is only weakly allowed 
the transition T|— *S is slow, and may persist even after the 
illumination has ceased. 



Phosphorescence involves a change of "multiplicity {an 
unpairing of spins) at an intermediate step, and this is its 
mechanistic difference from "fluorescence. From this point of 
view it follows that phosphorescence may occur if there is a 
suitable triplet state in the vicinity of the excited singlet states 
of the molecule and if there is a sufficiently strong spin-orbit 
coupling to induce ISC: a heavy atom enhances this crossing 
probability {the heavy-atom effect ) . Furthermore, there must 
be enough time for the molecule to cross from one curve to 
the other, and this means that the vibrational deactivation 
must not proceed so fast that the molecule is quenched and 
taken below the point where the curves intersect before the 
ISC interaction has time to operate. It is for this reason that 
many molecules which fluoresce in fluid solution are found to 
phosphoresce when they are trapped in a solid lattice, such as 
a gel or glass. We can also predict from Fig. P6 that the wave- 
length of the emitted light should be longer (further into the 
red) than fluorescent emission: the lowest vibrational level of 
the triplet lies below that of the lowest excited singlet. Finally, 
it is not impossible for some molecules to clamber back into 
the singlet Si, and to fluoresce into the ground singlet: this is 
stow fluorescence, the triplet state acting merely as a reservoir. 

What is the evidence that the triplet state is involved in 
phosphorescence? The first direct evidence came from the 
determination of the "magnetic susceptibility of a phosphor- 
escent sample with and without illumination: it became 
paramagnetic when the light was on. The most sensitive 
procedure is to apply "electron spin resonance to the phos- 
phorescent state: this shows unequivocally that a triplet 
state is involved. 

Questions. How does phosphorescence differ observational ly 
from fluorescence? Sketch the diagram corresponding to the 
formation of a fluorescent state and superimpose it on the 
diagram that leads to phosphorescence. What competitions 
determine which path is taken? How may phosphorescence be 
enhanced relative to the fluorescent and radiationless decay 
routes? What is meant by the term 'inter -system crossing', and 
how does it differ from 'internal conversion'? Why is phos- 
phorescence a slow process? What perturbation is responsible 
for the ISC and the emission? How may this perturbation be 
enhanced? What is slow fluorescence, and how may it be 



176 



photoelectric effect 



distinguished from phosphorescence? How may it be dis- 
tinguished theoretically and experimentally from delayed 
"fluorescence? What evidence is there that the triplet state is 
involved in phosphorescence? Discuss the reason why it is 
appropriate to think of the ISC as occuring at the intersection 
of the two potential curves. 

Further information. See MQM Chapter 10, "fluorescence 
and -laser action. See also Bo wen (1946), Wayne (1970), 
Calvert and Pitts (1966), and McGlynn, Azumi, and Kinoshita 
(1969). The electron spin resonance evidence for the role of 
the triplet state will be found in Hutchison and Mangum 
(1958) and Carrington and McLachlan (1967), and is reviewed 
in detail in McGlynn eta/. (1969). 

photoelectric effect. When short-wavelength light falls on a 

metal surface electrons are emitted. Three observations are 

very important: 

{1} the emission occurs only if the wavelength of the incident 

light is smaller than a threshold value characteristic of the 

metal; 

(2) emission occurs even at very low intensities so long as the 
threshold frequency is exceeded, and however dim the light 
there is no time-delay between its application and the appear- 
ance of photoelectrons; 

(3) the kinetic energy of the emitted electrons depends 
linearly on the frequency of the light once the threshold is 
exceeded. 

The first observation suggests that the energy that can be 
transferred to the meta! surface in order to eject an electron 
is related to the frequency of the light, and that the metal 
surface cannot gradually accumulate energy until it has 
sufficient for the job. This behaviour is non-classical, because 
an incident wave would be expected to deposit its energy 
into the metal irrespective of its frequency. 

The quantum theory and its concept of "photons can 
explain all the features of the effect in a simple and direct 
fashion. It recognizes that a light wave of frequency V may be 
considered as composed of a collection of photons each 
bearing the energy hv. The explanation of the three obser- 
vations is then as follows. 



1. When a photon strikes the metal's surface it can eject an 
electron by imparting all its energy to it, but the ejection will 
be successful only if the energy transferred hv is sufficient to 
overcome the energy that binds the electron to the metal (the 
"work function <p). If the frequency is less than the work 
function the electron will not be emitted, and the photon re- 
emerges from the surface as part of the reflected beam. Thus 
the threshold frequency of the photoelectric effect can be 
understood. 

2. The intensity and time characteristics are explained on 
the same basis, because so long as the frequency exceeds 
threshold the photon is able to eject an electron; the effect 
depends on single photon-electron collision events rather than 
the accumulation of energy from a passing wavefront . At low 
intensities {few photons) only a few collisions occur, but each 
photon carries the same energy hu as the photons in a heavily 
populated intense beam of the same frequency. 

3. The third point can be explained on the grounds that a 
successful photon is annihilated in the collision that ejects 
the electron, and, by the conservation of energy, all its 
energy must appear in the electron; of this energy an 
amount corresponding to the work function is expended in 
prising the electron out of the metal and setting it in free space, 
and the remainder hv — <p must be ascribed to the kinetic 
energy of the electron ?r> B v z . It follows that the kinetic 
energy of the electron is proportional to the frequency of the 
incident light. 

The importance of the photoelectric effect, other than its 
technological value for light-sensitive devices, lies in its 
historical value in the development of the idea that radiation 
is "quantized and in its application to the study of the "work 
function, for the latter can be determined from the threshold 
frequency. A modern development of this is photoetectron 
spectroscopy (PES), where electrons are ejected from molecules 
by a high-energy photon {from a short-wavelength source). The 
'work function' in this case is the energy required to extract 
the electron from the -orbital it occupies in the molecule, and 
so the appearance of electrons with a variety of different 
kinetic energies under the influence of monochromatic 
radiation implies that they are being ejected from a corre- 
sponding range of orbitals of various binding energies 
{Fig. P7). Analysis of the kinetic-energy spectrum of the 



photons 



177 




FIG, P7. Formation of a photoelectron spectrum. 

ejected electrons provides detailed information about the 
energy levels of molecules. When the photon source is in the 
X-ray region each photon carries sufficient energy to eject 
electrons from the innermost shells of atoms, and so their 
energies can also be studied (Fig. P8). This X-ray technique is 
known by the inglorious name of electron spectroscopy for 
chemical analysis (ESC A). See also "Auger effect. 

Questions. State the significant features of the photoelectric 
effect. Why cannot the classical wave theory of light account 
for these phenomena? What is the role of photons in the 
photoelectric effect, and how do they enable the features of 
the effect to be explained? What is the role of the work func- 
tion? Light of wavelength 750 nm, 500 nm, 200 nm falls on 
a surface composed of one of the following metals: Na (2'3 
eV), K (2-2 eV), Cs (2-1 eV), W (4-5 eV); the number in 
brackets is the work function for the metal. Predict the 



O K tt (54keV) 

Al K a (|.49 IteV)— 
Y M 5 CU2eV) 



He li(4leV) 
-He IC2I«V) 




FIG. P8. Ejection of electrons from different regions of an atom. 

kinetic energy of the ejected electron in each case where 
photoe mission may occur. The kinetic energy of photo- 
electrons as a function of incident frequency was measured: at 
a wavelength of 625 nm the kinetic energy was 0-2 eV, at 
416 nm it was 1-2eV, and at 312 nm it was 2-2 eV, Calculate 
the work function for the metal. What is the relation of 
photoelectron spectroscopy to the photoelectric effect? 

Further information. Analysis of the photoelectric effect will 
be found in Chapter 2 of Bohm (1951) and 81.3 of Jammer 
(1966). The implications of the phenomenon for the concept 
of photons is described by Born and Beim (1968). Photo- 
electron spectroscopy is described in Turner, Baker, and 
Brundle (1970), Baker and Betteridge (1972), and Siegbahn 
etal. (1969). Siegbahn (1973) has given a simple introduction 
to ESCA and Turner (1968) a simple introduction to PES. 

photons. Light of frequency V can impart energy only in dis- 
crete amounts ("quanta) of magnitude hv, where h is Planck's 
constant. Alight beam of frequency v therefore can possess an 
energy that is an integral multiple of hv, and so it is natural to 



178 



polarizability 



imagine this successive excitation of a frequency mode as an 
addition of hypothetical particles to a state. Thus a beam of 
frequency fand energy nh V could be regarded as containing n 
light corpuscles. These quanta of excitation, or hypothetical 
corpuscles, are photons. Each photon of wavelength A = civ 
carries an energy hv and, according to the *de Brogiie relation, 
a linear momentum hvlc. Low-frequency photons carry little 
energy and momentum; high-frequency photons carry much 
of both. 

Does light exhibit any of the corpuscular features that this 
photon description suggests? Indeed it does, for the "photo- 
electric effect confirms that energy can be transferred only in 
discrete amounts (corresponding to the annihilation of a 
photon), and the "Compton effect shows that each photon 
carries a characteristic amount of momentum related to its 
frequency. Radiation pressure can be understood in terms of 
the pressure imparted to a surface by a steady stream of 
incident photons. 

Photons may be polarized so that their electric component 
lies in a plane (plane-polarized light) or moves in a circle or 
ellipse; but in all cases the electric component must be 
perpendicular to the propagation direction. The photon 
possesses an intrinsic angular momentum (its "spin), and of 
this the existence of left and right circularly polarized light 
is a manifestation. The spin of a photon is unity: one pro- 
jection (that corresponding to a right-hand screw along the 
direction of travel) corresponds to left circularly polarized 
light, one projection (the left-hand screw) to right circularly 
polarized light, and the third (remember that for "spin 1 
there are three possible components on a selected axis) is 
forbidden to particles moving with the speed of light, which 
photons do. 

The existence of the spin of a photon is the basis of the 
^selection rules for "electric dipole transitions. Since its spin 
is unity the photon is a "boson: this entails, through the 
•Pauli principle, that an indefinite number may occupy a 
single quantum state, and for this reason intense mono- 
chromatic beams may be prepared {see "laser). 

Questions. What is meant by the term "photon? What is the 
energy of a photon in light of frequency v, and what 
momentum does it carry? What is the energy of the photon 



of light corresponding to the microwave region of the 
spectrum (1 cm"') and infrared (500 enf '), the visible 
(~55Q nm), the ultraviolet (300 nm), and 7-rays (10"" m)? 
How many photons are emitted in a 1 mJ pulse from a 337 
nm wavelength nitrogen-gas laser? How many photons are 
emitted from a tungsten 100 W lamp each second: assume all 
the radiation is at 450 nm? What frequency of light is the 
minimum that can be used to fracture a 100 kJ mof 1 bond 
in a molecule? What is the evidence for the corpuscular 
nature of light? Why is this view compatible with the exist- 
ence of the typically wave phenomenon of diffraction? 

Further information. The quantum mechanics of photons 
can be made very complicated; but it is very important. now 
that lasers are ascendant. If you are really interested see 
Grandy (1970), Louisell (1973), Berestetskii, Lifshitz, and 
Pitaevskii (1971), Akhiezer and Berestetskii (1965), Levich, 
Mayamlin, and Vdovin (1973), Jauch and Rohrlich (1955), 
and Kaempffer (1965). The -uncertainty relation for the 
phase of photons and their number is discussed correctly by 
Carruthers and Nieto (1968), but rarely correctly elsewhere. 
The study of photons is helpful in the discussion of the inter- 
pretation of quantum mechanics: see Dirac (1958) and 
Feynman, Letghton, and Sands (1963). Quantum optics is 
the study of optics where the quantum, and therefore the 
photon, aspects are dominant. See Glauber (1969) for a 
review and Loudon (1974) and Louisell (1973) for a modern 
account. Are photons particles? See Born and Beim (1968) for 
a view. See p. 9 of Whittaker's Stereochemistry and mechanism 
(OCS 5) for a helpful picture of the decomposition of a plane 
wave into its circular components. 

polarizability. When an electric field is applied to an 
individual atom or molecule the electron distribution and the 
molecular geometry are distorted; the polarizability is a 
measure of the ease with which this occurs. The atomic 
polarizability is the contribution to the overall polarizability 
due to the geometrical distortion. It is usually significantly 
smaller than the electronic polarizability , which is the con- 
tribution due to the displacement of the electrons. There is a third 
contribution to the polarizability of a bulk sample: this is the 
orientation polarization. It arises when the molecules have a 



polarizability 



179 



BOX 17: Polarizability 

Energy of potarizable molecules in a field E 

£(E)=£<°> -// >E- ^ -lj.^3 + . . . 
Dipole moment in a field E 

M is the permanent dipole moment, or the polarizability, 
and the first hyperpolarizability. (For magnetic properties, 
a is called the magnetizability, and written £.) 
Quantum expressions 

a = -(<y£/9E% s0 = § £Vd ,,d n0 . 
A„ = f^ 0) — E< 0) , the molecular excitation energy, 
d on = ^ ldli^ n ), the transition dipole moment. 
Polarization of medium 
Electric susceptibility ()^): P = e % E. 

Relative permittivity {dielectric constant): € = t + v. 

Refractive index: n = e /a 
r r 

Lorentz local field: E |qc - E + P/3e = |(e + 2)E 

(the factor 3 is approximate). 

Lorentz- Lorenz or Clausius-Mossotti equation: 



( e r 1 \ (m\ _ La 



M is the molecular weight, p is the density, and L is 
Avogadro's number. La/3e is referred to as the molar 
polarizability a 



M' 



Debye equation (for molar polarizability of polar 
dielectrics): 

a A : atomic polarizability 



a E : electronic (molecular) polarizability 

a Q : orientation polarizability, which is given by the 
Langevin contribution for a molecule with 
permanent dipole moment uS a \ as 

0L o =l± W2 l3kT, 
See Table 20 for the Maxwell equations. 



permanent dipole moment: the applied field orientates the 
molecules, and the entire sample acquires net polarization. The 
orientation effect is not complete because thermal motion 
disorganizes the sample: its magnitude may be calculated by 
applying the Boltzmann distribution to determine the mean 
dipole moment of the sample, and one deduces the Langevin 
contribution exhibited in Box 17. This mechanical con- 
tribution will not concern us further (although it is an 
important component of the total polarization); we shall 
concentrate on the source of the electronic contribution. 

Highly polarizable molecules respond strongly to the 
application of the field; they become highly polarized, and the 
centroid of negative electronic charge is displaced. If the mole- 
cule is initially nonpolar the polarization leads to the form- 
ation of an electric "dipole moment; and if it is already polar 
it leads to an additional component of the dipole. The 
magnitude of the induced dipole is a good indication of the 
polarizability of the molecule, and the polarizability a may 
be defined as the constant of proportionality between the 
induced moment and the strength of the electric field: 
u(induced) =aE. The dipole moment mighfdepart from this 
linear relation if the applied field is very strong; in that case 
the dipole depends on E 2 , and higher powers, and the coef- 
ficients of E 2 , F 3 , . , . are known as the first, second, . . . 
hyperpolarizabilities. We shall neglect these non-linear response 
terms and concentrate on the linear response, the polarizability 
a. 

The quantum-mechanical calculation of the polarizability 
proceeds by calculating the energy of a polarizable molecule in 
an electric field, and relating this to a second-order 
"perturbation-theory calculation (see Box 17). It is found that 
the magnitude of polarizability can be interpreted in a variety 
of ways. 



180 



polaron 



1. One interpretation shows that the polarizability increases 
with the size of the atom and with the number of electrons it 
contains; this can be understood in terms of it being easier for 

a field to distort the electronic distribution when the electrons 
are far from the nucleus, or well shielded from its charge. 

2. Another interpretation, which is based on the view that 
the distortion of the molecule can be represented by an 
appropriate "superposition of wavefunctions, gives the 
expression for the polarizability as a sum of terms, each one of 
which represents an admixture of an excited state into the 
ground state. The amount of each state depends on its energy 
separation from the ground state and on the intensity of an 
optical, -electric dipole transition between it and the ground 
State: as the intensity of the transition increases the state may 
contribute more strongly, and as the energy increases it con- 
tributes less. A consequence is that it is reasonable to expect 
molecules that have intense transitions in the optical or lower- 
frequency region of the spectrum to be highly polarizable. On 
this basts it is understandable that the aliphatic hydrocarbons, 
which have weak optical transitions in the ultraviolet, are only 
weakly polarizable. 

3. A third interpretation remarks that the polarizability 
formula can be understood in terms of the magnitude of the 
fluctuations in the instantaneous electric dipole moment of 
the species. It is possible to imagine local transient electronic 
movements in the molecule that give rise to a dipole moment 
which on the average is zero (for a non polar molecule); the 
greater these fluctuations the greater the polarizability. This 
interpretation is related to the first, because the fluctuations 
are greater in large, weakly-bound systems. 

The polarizability of a sample is frequency-dependent. At 
low frequencies (below about 10 12 Hz) the molecules, the 
atoms within the molecules, and the electrons of the molecules 
can follow the changing direction of the applied field. At 
higher frequencies (above 10 12 Hz but below 10 14 Hz) the 
molecules cannot reorientate themselves sufficiently quickly, 
and so the orientation polarizability ceases to contribute. At 
still higher frequencies the atomic nuclei are too sluggish to 
follow the oscillating field, and the atomic contribution is 
eliminated. This implies that at high frequencies (in the 
optical range) the only contribution to the polarizability is the 
electronic, but even this ceases at very high frequencies. The 



frequency-dependence (the dispersion) of the polarizability is 
a helpful property in distinguishing the contributions. 

Polarizabilities of molecules are related to the relative 
permittivity (dielectric constant) of the medium they compose 
{see Box 17), and its "refractive index. Both properties enable 
the polarizability to be determined over a wide frequency 
range. (Since the refractive index is normally measured at 
optical frequencies it is related to the electronic polarizability.) 

Questions. 1. What happens when a molecule is immersed in 
an electric field? How may the polarizability be defined? What 
is the dependence of the induced dipole moment on the 
applied field when the latter is very strong? Would you expect 
the polarizability of a molecule to depend on its orientation 
with respect to the field? Why should the polarizability 
increase as the species becomes larger and its electrons less 
strongly bound? Which is more polarizable, He or He + ? What 
is the role of the instantaneous fluctuations of the dipole 
moment of an atom in determining its polarizability? Can the 
polarizability be related to the strength and frequency of 
optical transitions? Why does the polarizability depend on the 
admixture of excited states? {Revise -perturbation theory and 
■virtual transitions.) 

2. Apply the expression in Box 17 to calculate the polariz- 
ability of a charged simple -harmonic oscillator, an electron in 
a one-dimensional square well, and the -hydrogen atom. Where 
you feel it necessary apply the -closure approximation. Cal- 
culate the polarizability parallel and perpendicular to the axis 
of a 2p -orbital in the carbon atom: use "Slater atomic 
orbitals and the closure approximation. 

Further information. See MQM Chapter 1 1 for a derivation of 
the relations in the Box and for a further discussion. See also 
van Vleck (1932), Davies (1967), and Buckingham {1960). 
Polarizabilities are listed in Landolt-Bornstein. For further 
applications see -intermolecular forces and "dispersion 
forces. Hyperpolarizabilities and their measurement are dis- 
cussed by Buckingham and Orr (1967). See also Kielich 
(1970). 

polaron. A polaron is a defect in an ionic crystal that is 
formed when an excess of charge at a particular point 
polarizes the lattice in its vicinity. Thus if an electron is 



precession 



181 



OOOOCDOO 
OOOOCD0O 

eeeee 

eeee 

ooo 

eeee 

eeee 




cpoooc 
ooooc 



eeee^ 

ee^ 

oooc 

»eee£ 



OOOOMOWOOOC 

000000000000c 



FIG. P9. A polaron in a simple lattice. The doubly hatched site is a 
doublv charged region; the singly hatched sites are singly charged 
with opposite sign. 

captured by a halide ton in an alkali-halide crystal the metal 
ions move towards it and the other negative ions shrink away 
(Fig. P9). As the electron moves through the lattice it is 
accompanied by this distortion. Dragging this distortion around 
effectively makes the electron into a more massive particle, 
and this is intended to be implied by the name polaron: a 
lattice distortion moving through the lattice like a massive 
particle. 

Further information. See Chapter 10 of Kittel (1971) and 
Kittel (1963). A good discussion is given in Chapter 4 of Mott 
and Davis (1971 ), and the subject is reviewed in Kuper and 
Whitfield (1963). 

precession. In the -vector model of the atom and in the 
general theory of -angular momentum, an angular momentum 
of magnitude [/{/ + 1)] y 'h andz-component/nfi is represented 
by a vector of length [/"(/ + 1 )] K making a projection m on to 
az-axis. From the -uncertainty principle it is known that if the 
^-component of angular momentum is precisely specified then 
the x- and the y-components are completely uncertain. This 
situation may be represented in the vector diagram by 
indicating the range of possible orientations of the angular 




FIG. P10. Precession. The cone denotes the possible but indeterminate 
orientations of the vector. 



momentum by a cone (Fig. P1Q, but see also Figs. A2 and A3): 
wherever the vector lies on this cone, it has the samez- 
component but its*- and y-components are completely 
undetermined. The cone is referred to as the cone of 
precession. Note that we avoided saying that the angular 
momentum actively precesses around z: the cone represents 
the possible array of orientations of the angular momentum, 
and at this stage we do not wish to imply that the tip of the 
vector moves round the mouth of the cone. In the absence of 
magnetic fields the vector is at rest at an indeterminate position 
on the conical surface. 

When a magnetic field ts applied along the z-axis the states 
with different projection m have different energies by virtue of 
the magnetic moment associated with the angular momentum: 
the energy of the state m ism^ a S for orbital angular 
momentum (and 2m(i B B for »spin}. This energy can be 
expressed as a frequency by dividing by h: CU= mft B/ti The 
vector diagram is a symbolic code representing the quantum 
mechanics of the situation, and it can be augmented in a way 
that incorporates the energy of the state by adopting the 
convention that a state with energy mp. B is represented by a 
vector that revolves around the z-axis with a frequency 03 
(Fig. P1 1 ) : this is precession, and the frequency is the Larmor 
precession frequency. As the field is made weaker the 
precession frequency slows, and in the limit of zero field the 
static, indeterminate distribition of vecots is regained. The 
state with the greatest value of \tn I precesses most rapidly, and 



182 



precession 



fairer 



still 



falter 




=/\B/fi 



cj-0 



-Pffr 



FIG, Pit. Larmor preeBssion in the presence of a magnetic field, and its 
connexion with the vector model, 

that with zerom precesses not at all; opposite signs of m are 
interpreted as opposite senses of precession. 

From this picture we see that the rate of precession about 
an axis represents the strength of coupling to that axis, and 
this view may be extended to situations where the energy 
arises from sources other than external magnetic fields. The 





FIG. P12. Two points of view about the precession of coupled vectors. 
In (a) we sit on I and observe s and j; in (bl we sit on j and observe 5 
and I. 



case of °spin-orbit coupling is an example: I and s are coupled 
by the spin-orbit interaction, and form a vector j. If the spin- 
orbit coupling is strong the vector-coupling scheme involves a 
rapid precession of s about 1 (or, what is equivalent, a rapid 
precession of 1 and s about their resultant j): this is illustrated 
in Fig. P12. When the coupling is weak the precession is slow 
and the coupling can easily be broken by other influences. 

The picture of precession is a superb example of the way 
that algebraic concepts of quantum mechanics can be 
represented by diagrams: each aspect of the 'picture' is a code 
for some aspect of the quantum-mechanical situation. But 
then, you might think, is the quantum-mechanical algebra 
not itself merely a code? 

Questions. What does the cone of precession represent in the 
vector model of angular momentum when no field is present? 
Why is a cone forced on the model? What does it represent 
when a magnetic field is applied? Upon what does the 
precession frequency depend? Calculate the Larmor 
precession frequencies for a p-electron in a magnetic field of 
1 G and 3 kG; what is the Larmor frequency for an electron 
-spin in a field of 34 kG and of a proton spin in a field of 
15 kG? What in general does the rate of precession represent? 
Is it easy to decouple two momenta that are rapidly precessing 
about each other? The spin-orbit coupling energy of an elec- 
tron in a first-row atom is about 50 cm -1 : compute the 
relative precession frequency for this situation, and estimate 
the external magnetic field that would be required to cause a 
significant decoupling of the momenta and a significant 
orientation of the individual momenta with the field. The 
conversion of a triplet state into a "singlet can be brought 
about by changing the relative phase of the two electron-spin 
orientations: thus if one spin precesses faster than the other 
its orientation is shifted by 180° with respect to the other, and 
a singlet is generated out of a triplet (see -singlets and triplets, 
especially Fig. S7). Suppose that the two electrons of a triplet 
molecule are in fields differing by 1 G, either by virtue of 
inhomogeneities in the field or because they experience 
different internal fields. Calculate, by determining the different 
Larmor precession frequencies, the rate at which the singlet 
is formed. Can you see a connexion with the interconversion 
of ortho- and para-hydrogen (see "nuclear statistics)? 



predissociation 



183 



Further information. See MQM Chapters 6 and 8 for a dis- 
cussion of the algebra beneath the picture, and the applications 
of the "vector model to atomic and molecular spectra. See 
also Candler (1964), Herzberg (1944), White (1934}, and Kuhn 
(1962) for applications in atomic spectroscopy. The language of 
precession is used extensively in the discussion of magnetic- 
resonance experiments; therefore see McLaucfolan's Magnetic 
resonance (OCS 1), Lynden-Bell and Harris (1969), Carrington 
and McLachlan (1967), and Slichter (1963). A good example 
of the decoupling of two precessing vectors is provided by the 
°Paschen-Back effect. 

predissociation. Ordinary, well-behaved dissociation occurs 
when a molecule is excited to a state that possesses more 
energy than the separated fragments, A transition from curve 
X to curve A in Fig. P13 a provides such an example; the 
spectrum is blurred where the upper vibrational states are not 
•quantized (where they are really translations! states}. Pre- 
dissociation is dissociation that occurs in a transition before 
the dissociation limit is attained, hence its name. 



them to remain until enough energy is added to excite'the 
molecule beyond the dissociation limit of the upper state. In 
Fig. P13 b we see that the vibrational structure disappears at 
this point. (Below this point it has the intensities characteristic 
of the "selection rules in operation and the °Franck : Condon 
principle.} In some cases it is observed that the vibrational 
structure of the spectrum disappears before the dissociation 
limit is attained. This may occur because another, dissociative 
state (B) crosses A, and because there exists an interaction 
("perturbation) in the molecule that can flip the state of the 
excited molecule from A to B (Fig. P13 c). Put another way, 
this can be interpreted as the states in the vicinity of the 
region A in Fig, P13 c being a mixture: the vibrational states of 
A acquire some of the translational character of the states of B 
at that energy. Therefore a state within the region A has some 
propensity to dissociate even though its energy may be far 
below the dissociation energy for the state A. When the energy 
of the incident light is great enough to excite the vibrational 
state of A above the region A the lines are again observed to be 
sharp, for now the molecule is unable to switch into the state 




dissociation limit 





FIG. P13. (a) Dissociation by a transition to a dissociative state, (b) Dissociation when enough energy is added to disrupt the upper state, (c) Pre- 
dissociation due to the presence of a dissociative state. The appearances of the spectra are indicated on the left. 



Consider the transitions A *— X shown in Fig. P13 b. On 
the -electronic transition is superimposed a series of lines due 
to the transition to the vibrational states of A. At low 
frequencies (energies) the lines are sharp, and so we expect 



B, The presence of predissociation may be recognized therefore 
by a blurred region in the vibrational -progression of an elec- 
tronic transition. 

The mechanism of predissociation is closely related to the 



184 



progression 



•Auger effect, the principal difference being that the former 
dissociates a fragment of the molecule and the latter spits out 
an electron. In each there is a radiationless transition from a 
bound to an unbound state. Predissociation obeys certain 
•selection rules, and these will be found listed in that section. 

Induced predissociation is predissociation that is induced 
by some external influence; in particular, collisions with a 
foreign gas {'collision-induced predissociation') or an applied 
field. The collisions are able to knock the excited molecule 
from A to B, and the applied field may relax some of the 
selection rules that govern the A-vWV* B transition (the wavy 
line denotes that the transition is radiationless). 

Questions. 1. What is meant by the term 'predissociation'? 
How may it be recognized in a spectrum? Suppose the 
emission of light from a molecule were observed: what effect 
would predissociation have on the appearance of the emission 
spectrum? What causes the molecule to flip from one curve 
to another? How may the efficiency of this transition be 
enhanced? In what senses is predissociation a type of "Auger 
effect? What happens to the rotational structure of the 
electronic transition in the region close to the predissociation 
domain? Suppose that the state dissociates with first-order 
kinetics and lifetime r, use the "uncertainty principle to find 
an expression in terms of r which determines whether or not 
rotational structure will disappear from the spectrum, and then 
find an expression for the disappearance of vibrational struc- 
ture. Choose typical values of the molecular properties you 
require, and assess the lifetimes of levels for which the 
rotational and vibrational structures disappear. 
2. In the text it was remarked that outside the region A 
there was only insignificant mixing of the states A and B. On 
what grounds may that statement be justified? Provide a 
classical and a quantum-mechanical interpretation; for the 
latter consider the role of overlap in the same way as in the 
justification of the "Franck-Condon principle. 

Further information. See MQM Chapter 10, See Herzberg 
[1950, 1966) for a thorough discussion of predissociation and 
many examples. See also Barrow (1962), Gaydon (1968), and 
King (1964). Induced predissociation is described by Wayne 
(1970). 



progression. In the "electronic spectra of molecules the 
excitation of the electron is accompanied by excitation of the 
"vibrations, and so instead of a single line in the spectrum there 
may be a complicated band of transitions. A progression is a 
series of lines that arise from transitions from the same 
vibrational level of one of the states (the ground electronic 
state if absorption is involved) to successive vibrational levels 
of the other state. Thus the v" = progression is the series of 
transitions starting in the v" = vibrational level of the ground 
electronic state of the molecule and terminating in the V = 0, 
1,2,... vibrational levels of the upper electronic state (Fig. 
P14). The v" = 1 progression is a similar series starting in the 
(/' = 1 vibrational level of the ground electronic state. The 
lines in a progression are labelled (v' , v"); therefore the 
if = progression consists of the transitions (0, 0), (1, 0), 
{2, 0), etc.. 



+ 


■ , 




J 






.' 










I 








J 


, 



















































































C0,0KI,0>«,Q> 



5 U 
4 

3 
2 

I 



6 u" 
5 

4 

3 



FIG. P14. A progression. 



Questions. What is a progression? How does it differ from a 
"sequence? What information can you extract from the 
positions of the lines in a progression, and what information 
can you obtain fay comparing the v" = and the v" = 1 
progressions? Would you expect the intensities of the pro- 
gressions to be the same? Given that the energy of a 
vibrational level of the ground state depends on v" through the 



progression 185 

expansion iJ'{u" + ~) + x"oJ'{if + |-) 2 , with a similar 
expression for the upper state, calculate the frequencies of the 
if — and the v = 1 progressions. 

Further information. See MQM Chapter 10. Detailed 
information about the appearance, analysis, and formation 
of progressions will be found in Barrow (1962), Whiffen 
(1972), Gaydon (1968), King (1964), and Herzberg (1950). 



Q 



quadru poles and other multipoles. The electric quadrupole 
moment is one of the series of rnultipole moments which are 
used to describe the way electric charge is distributed over a 
body. The first member of the series is the electric monopole, 
more commonly referred to as the point charge; then comes 
the electric 'dipole moment, which may be regarded as the 
juxtaposition of two opposite charges, and so has no net charge 
(no monopole moment); then comes the electric quadrupole 
which may be regarded as being formed from four charges 
arranged in a way that leads neither to net charge (no monopole 




octopole 
FIG. Q1 . Various electric rnu I tipoles. 



hexodecapole 



moment) nor net dipole moment. An arrangement that 
achieves this is shown in Fig. Q1 . Higher multipoles may be 
constructed in an analogous way: for a 2 n -pole {n = 1 is a 
dipole, n - 2 a quadrupole, n = 3 an octopole, n = 4 a 
hexadecapole) it is necessary to arrange 2" electric charges in 
an array that possesses no lower rnultipole. Some of these are 
represented in Fig. Q1, but alternative arrangements may also 
be envisaged. 

Often many multipoles vanish by virtue of the symmetry of 
the molecule. Consider, for example, the molecule CO2 which 
has the linear structure 0=C=0. As the molecule has no net 
charge, it has no electric monopole moment. It is symmetrical 
about the carbon atom, and so it has no electric dipole moment. 
The oxygen atoms are more "electronegative than the carbon, 
and so they bear a higher charge density than the central 
carbon; therefore the charge distribution has the form 
(5 —)—{S + 8 +)-(S -). This is of the form of a quadrupole, 
and so we can expect the quadrupole moment of CO^ to be 
non-zero, as indeed is found. 

The potential arising from a general charge distribution may 
be represented by a sum of potentials of the multipoles into 
which the distribution may be divided. The electric potential 
due to a 2 n -pole falls off with distance according to 1/r" +l ; 
therefore the electric field due to a 2" -pole falls off 
according to l/r n * 2 (the field is the negative gradient of the 
potential). It follows that the higher multipoles have a much 
shorter range than the lower: a monopole potential falls off 
according to the Coulomb law 1 lr, a quadrupole according to 
I// 3 . The more rapid decay of potential in the higher 
multipoles may be understood in terms of the way that a 



186 



cluster of electric charges (as depicted in Fig, Q1 ), when 
viewed from great distance, is hardly distinguishable from 
having nothing at all: the cancellation at large distances occurs 
more effectively the higher the rank of the multipole. 



quadruples and other multipoles 



187 



BOX 18: Multipole fields 
Field at r due to a point multipole at origin 



£ n *1/r"* a . 
Potential at r 

<x Mr"* 1 

dip0le: \4neJ {?) cos B <9. -^ separated by R, n = qR). 

linear quadruple: j(ff^) (^J (3cos 2 0-1) 

(quadrupole formed with ju, fi head-to-head, eQ = 2o7? 2 ). 
Energy of multipole in a field E 

(1) point charge (monopoie) £ = q(p[r), where 

(2) dipole: &<•*-%£ 



(3) quadrupole: £ = $eQ{d 2 <t>/dz 2 )[3 cos 2 0-1). 



J 



Different multipoles interact with different features of the 
electric field. An electric monopote interacts with the electric 
potential itself (see Box 18). An electric dipole interacts not 
with the potential but with its gradient; that is, the electric 
dipole interacts with the electric field. This can be understood 
in terms of the structure of the dipole as two juxtaposed 
point charges: at a general orientation of the dipole one charge 
interacts with the potential at its position and the other 
opposite charge interacts with the potential at its position. 
Only if the potential is different at the two points is there a 
net interaction; therefore there is an interaction only if the 
gradient of the potential does not vanish at the position of 
the dipole. Those to whom extension of analogies gives 



pleasure will be pleased to discover that the electric quad- 
mpole moment interacts with the second derivative of the 
electric potential; or, what is the same thing, with the gradient 
of the electric field. This may be understood by considering 
the situations depicted in Fig. Q2. In the first, where the 
gradient of the potential (the field) is constant over the 
quadrupole, the energy of interaction is independent of 
orientation, and so there is no net interaction. In the second, 
where the potential has curvature (and where the field has a 
gradient), the energy depends on the orientation, and 
therefore we conclude that there is an interaction. The 
mathematical form of the interaction is given in Box 18. 




r^) 



>f 



f-f-'-f-} 



.'field 
/ 

FIG. Q2. (a) A quadmpole in a constant field has an energy indepen- 
dent of orientation, (b) A quadmpole in a field gradient has an energy 
dependent on orientation. In each case the density of shading denotes 
the strength of ihe potential. 

The typical magnitude of a molecular mono pole is of the 
order of the magnitude of the electronic charge e (1.6 X 10" 19 C). 
A single charged ion may be regarded, by the pedantic or the 
neat, as a monopofe of this magnitude. The magnitude of a 
molecular dipole is of the order of eR, where R is a molecular 
diameter; for R - 0.1 nm (1 A) the order of magnitude is IfX 29 



188 



quantum 



C m (but see °dipole moment for debye). In the same way we 
may estimate the order of magnitude of the electric quadrupole 
moment of a molecule as eR 1 , or roughly 10" 3 * C m 2 . Part of 
the absurdity of the size of the unit is eliminated by defining 
the electric quadrupole moment as eQ, and calling Q the 
'quadrupole moment'. It is normally expressed in square 
centimetres; therefore a molecular quadrupole moment might 
be expected to be of the order of 10~ 16 cm 2 (1 A 2 ), and a 
nuclear quadrupole moment (see -hyperfine effects) of the 
order of 10" 24 cm 2 . 

Questions. 1. Draw diagrams of charge distributions that 
represent an electric quadrupole, octopole, hexadecapole, 
32-pole. Now draw alternative structures with similar 
multipoles. What is the dependence of distance of the 
potential due to a 2"-pole? What is the dependence on distance 
of the field due to a T -pole? Why is the electric quadrupole 
moment the most important multipole moment of COj? 
What is the first non-vanishing multipole moment of the 
helium atom? Of methane? Of water? Why does the potential 
of a quadrupole fall off more rapidly that that of a dipole? 
With what aspect of an electric field does a quadrupole interact? 
Why does a quadrupole not interact with a constant electric 
field? With what aspect of the field does an electric octopole 
interact? Draw a diagram to illustrate the physical basis of 
your answer. How may a molecular quadrupole be measured? 

2. Take the quadrupole arrays of charges illustrated in Fig. Q1 
and consider a point at a distance r from the centre of both 
arrays. Write an expression for the potential at that point due 
to the multipole (by regarding each point of the multipole as 
the source of a Coulomb field). Assume that r is much greater 
than the separation tfof charges within the multipole, and 
expand the potential in powers of d/r; retain the leading term 
in the expansion. In this way find the electric potential due to 
a quadrupole. Then find the field at the point r. Compare your 
answer with the expression in Box 18. 

3. Put a single electron at a distance r from the centre of a 
quadrupole of (a) molecular dimension and (b) nuclear 
dimension. Calculate the energy needed to rotate the quad- 
rupole through 90°. What size quantum would be needed to 
invert a nucleus of this nature? To what frequency does that 
correspond? This calculation is the basis of the technique of 



nuclear quadrupole resonance (n.q.r.), where electric field 
gradients within molecules are determined by observing the 
energy required to rotate the orientations of nuclei with 
electric quadrupole moments. 

Further information. Simple calculations on the electrostatics 
of multipole moments will be found in Corson and Lorrain 
(1970); tougher accounts are given by Jackson (1962) and Rose 
(1955, 1957). See Sugden and Kenney (1965) for a simple 
account with special attention paid to the way that quadrupole 
moments play a part in molecular spectroscopy. The calculation 
of molecular quadrupole moments is described by Davies (1967) 
and their measurement by Buckingham (1965), who also 
considers their rote in intermolecular forces. The determination 
of nuclear quadrupole moments is described by Lucken (1969), 
who also gives a detailed account of nuclear quadrupole 
resonance spectroscopy; see also Das and Hahn (1958). A 
recent review of n.q.r. is that by Chihara and Nakamara (1972). 
See Carrington and McLachlan (1967) for a simple account of 
the effects of quadrupole moments in magnetic resonance, 
and Chapter 6 of Slichter (1963) for a more sophisticated, but 
well stated, account. Tables of nuclear quadrupole moments 
are given in §8 of Gray (1972), and a few will be found in 
Table 17. 

quantum. A quantum of energy is the smallest amount of 
energy that may be transferred to or from a system. In 
classical physics there is no limitation to the smallness of 
permissible energy changes of a system, but quantum physics 
shows that only quantities of energy of a well-defined size 
can be transferred, the actual size depending on both the 
system and its state. A particularly simple example is provided 
by the -harmonic oscillator of natural period v. In classical 
theory the oscillator may swing at its natural frequency with 
some energy, and its energy may be changed continuously. 
Quantum theory shows that the oscillator's energy can change 
only by some integral multiple of hv, where h is Planck's 
constant. Thus an attempt to change its energy by a fraction 
of hv must fail. 

The magnitude of quanta involved in the behaviour of 
macroscopic objects is so small that the changes in energy are 
virtually continuous, but at a microscopic level the quan- 



quantum defect 



189 



tization cannot be ignored. For example, the size of the 
quantum needed to transfer energy to and from the pendulum 
of a grandfather clock {v~ 0.5 Hz) is only 3 X 10~ 34 J; but 
the oscillation of a bond in a molecule occurs at a frequency 
of the order of 10' 4 times larger at 6 X 10~ 20 J, or 36 kJ mol -1 

When an atom falls from an excited state of energy £ the 
change of energy f ( - E g is emitted as a quantum of light with 
a frequency V given by the Bohr frequency condition hv = 
E e - f g- The size of the emitted quantum depends on the 
states involved in the transition: quantum changes of large 
energy appear as high-frequency radiation, and those of small 
energy as low-frequency radiation. 

Planck's constant itself has the dimensions of action (J s) 
and so may be regarded as the fundamental quantum of action. 
The early approach to quantum theory proceeded by limiting 
the action of a system to multiples of h, and the 
Bohr-Sommerfeld quantization condition, which was the 
basis of the old quantum theory, was based on this 
quantization scheme. The dimensions of angular momentum 
are the same as those of action, and the natural quantum of 
angular momentum is also Pfanck's constant: the -angular 
momentum of a rotating body may be changed only in 
amounts of the order of h (h = h/2ir is a better order of 
magnitude), and so a rotating wheel cannot be decelerated 
continuously and smoothly, but must deliver up its angular 
momentum (and its energy) in steps. Once again the angular- 
momentum quanta are so minute for macroscopic bodies that 
it can be varied virtually continuously, but on an atomic scale 
the effects of quantization are of profound importance. 

Questions. What is a quantum of energy? Discuss the excitation 
of a simple pendulum in both classical and quantum terms. 
How is the energy of the quantum associated with the 
excitation of an oscillator of frequency Vl Calculate the size 
of the energy quantum for processes of natural frequencies 
1 Hz, t0 10 Hz, and 10 1S Hz. Calculate the energy of a 
quantum of red, yellow, and blue light, of ultraviolet light, 
and of X-rays (get help from Table 5). 

Further information. See MQM Chapter 1 for an historical 
account of the realization of the necessity of quantization, 
and Chapter 3 for some of its simpler manifestations. See 
Jammer (1966) for an historical perspective, and Hoffmann 



(1959), Heisenberg (1930), and Andrade e Silva and Lochak 
(1969) for gentle accounts. The experimental evidence is also 
reviewed In Moelwyn-Hughes (1961), Slater (1968), and Bohm 
(1951). See Feynman, Leighton, and Sands (1963) for a 
thorough and illuminating discussion of the whole subject. 
The Josephson effect provides an excellent method for 
measuring h\ for a description of the effect, see Langenberg, 
Scalapino, and Taylor (1966), and for its application, see 
Taylor, Langenberg, and Parker (1970). 



quantum defect. The spectrum of atomic "hydrogen 
(which is illustrated in Fig. H10 on p. 104) consists of 
several "series of lines which can be represented as the 
difference of two terms, both of which have the form 
Bin 1 , where R is the -Rydberg constant and n is an integer. 
The spectra of the alkati metals may also be grouped into 
series that can be expressed as the difference of two terms of 
the form R'/n 2 ; but on account of the repulsive effect of 
the core electrons the number n, the effective quantum 
number, is no longer an integer. It may be written as 
n — 5, where n is an integer and 5 is a correction called the 
quantum defect. This defect diminishes as the principal 
quantum number of the electron increases, for the electron 
is further away and the nucleus and its surrounding core 
electrons resemble quite accurately a single positive point 
charge, and the spectrum and the energies become more 
hydrogen-like. Since the quantum defect depends very 
strongly on the interaction of the valence electron and the 
core electrons it is not surprising that the greatest defects 
occur for s-orbitals, which "penetrate most closely to the 
nucleus. The quantum defect is a guide to the extent of 
penetration, but it has little other theoretical significance 
or importance. 

Further information . The quantum defect occupied more of 
the older literature than it does of the modern. Mention of it 
will be found in 83.6 of King (1964), §1.5 of Herzberg (1944), 
and Chapter III of Kuhn (1962), who lists some values and 
discusses why they were of importance. In this connexion, 
see §8.5 of Hartree (1957) and S8 of Condon and Shortley 
(1963). 



190 



quantum electrodynamics 



quantum electrodynamics. The apotheosis of present-day 
quantum mechanics seems to be quantum electrodynamics, 
although it is not entirely clear that the theory exists. What 
might exist is a unified view of particles and fields in which 
the electron is a manifestation of the electromagnetic field. 
Like Hamlet's ghost the electron momentarily slips back into 
the electromagnetic field, loses corporeality, and then 
regroups itself again as a recognizable particle. The electron 
spends 1/137 of its existence as radiation {which, incidentally, 
is about one-quarter of the proportion of the play Hamlet 
pere spends as ghost), and the remainder as particle (see 
■fine structure constant). This intimate connexion between 
matter and radiation is emphasized by quantum electro- 
dynamics, which treats the radiation field and the particle as 
the same object. One consequence of the electron being a 
manifestation of the electromagnetic field is that its motion 
cannot be smooth on a microscopic scale, but it should be 
envisaged as jitterbugging along: Zitterbewegung is the German 
with which the idea is elevated into respectability. 

For the chemist there are two important manifestations of 
this jitterbugging. One is the Lamb shift and the other is the 
free-spin "0-value of the electron. According to the "Dirac 
theory of the "hydrogen atom the levels 2 Syi and 2 P]/2 should 
be exactly degenerate, but Lamb measured a very small 
splitting (of the order of 1057 MHz). Therefore the Dirac 
theory, good as it is, must be wrong. 

In order to sketch the quantum-electrodynamic explanation 
of the Lamb shift we must first establish a crude picture of 
what is meant by the vacuum of the electromagnetic field. The 
picture of a light beam as a collection of "photons is based on 
the view that a mode of frequency V of the electromagnetic 
field can be excited to an energy which is an integral multiple 
of hv. This is analogous to the way in which a harmonic 
oscillator can be excited, and it is tempting to extend the 
analogy, and to say that, when no excitation is present (no 
photons), the mode still possesses a zero-point energy. This 
zero-point energy may be traced to the unquenchable zero- 
point fluctuations of the electric and magnetic fields, just as in 
a harmonic oscillator the motion cannot be entirely eliminated. 
The field oscillations buffet the electron, and so it jitterbugs 
about an equilibrium position. This slight smearing of the pos- 
ition of the electron affects the energy of an electron in an s- 



orbital slightly more than one in a p-orbital because in an s-orbital 
an electron goes closer to the nucleus and the smearing effect 
is more important, and the consequence is that their energies 
diverge. This shift may be calculated by quantum electro- 
dynamics, and almost exact agreement with experiment is 
obtained. 

The ^-value of an electron measures its effective "magnetic 
moment arising from its spin. Once again we may envisage an 
electron as being buffeted by zero-point electromagnetic 
fields as it spins, and, instead of spinning smoothly, it rocks. 
If we imagine a vibration in the equatorial plane the magnetic 
moment may be calculated, and once again essentially exact 
agreement with experiment is obtained. 

Further information. Books on quantum electrodynamics, 
disarmingly referred to as QED, are difficult to penetrate. 
See G randy (1970), Power (1964), Feynman {1962a, 19626), 
Roman (1969) (who gives a helpful guide to further reading), 
Bjorkenand Drell (1965), Henley and Thirring (1962), 
Thirring (1958), and Schweber (1961). A collection of 
significant original papers has been prepared by Sch winger 
(1958). The existence of zero-point fluctuations in the vacuum 
is also related to the attractive force that two conducting sheets 
exert on each other at small separations: for details of this 
Casimir-Polder interaction , see p. 142 of G randy (1970) and 
§3.4 of Power (1964). 

quantum numbers. Quantum numbers are labels that 

distinguish the state of a system and, in simple cases, enable 
the value of an observable to be calculated. Thus the state of 
a "particle in a box is labelled by the quantum number/?, which 
may take all integral values greater than zero, and a particle in 
a state labelled n has an energy n 2 {h 2 /SmL 2 ), The state of an 
electron in a "hydrogen atom is fully determined if we give the 
numerical values of the quantum numbers n, E, mg, s," 7 ^ n ' s 
The principal quantum number and determines the energy 
through £ = —/? H /n 2 {/? H being the -Rydberg constant); H the 
azimuthal quantum number, or the orbital angular momentum 
quantum number, determines the magnitude of the orbital 
angular momentum through the expression [2(fi + 1)] 'hand 
the number of possible orientations of this momentum through 
2£ + 1 ; m g selects which of the orientations the orbital angular 



quantum numbers 



191 



momentum does in fact have {and therefore which of the 
2fi + 1 atomic orbitals of given n and £ the electron occupies) 
and is called the magnetic Quantum number (because if a 
magnetic field is present the electron will have an energy 
myj± & B on account of its orbital -magnetic moment) ;s is the 
"spin quantum number which determines the magnitude of the 
spin angular momentum through [s{s + 1)1*6, and for an elec- 
tron s is fixed at the value s = j} m s , the magnetic spin quantum 
number, determines the orientation of the spin angular 
momentum relative to some axis, and if a magnetic field lies 
along this axis there will be an additional contribution to the 
energy of mj^S. 

For atoms other than hydrogen the orbital energy also 
depends on the value of £ on account of the effect of 
-penetration and shielding, but it is no longer possible to give a 
simple analytical connexion between the values of the quantum 
numbers and the energy of the state (but see -quantum defect). 



BOX 19: Quantum numbers 

Capital letters are used for quantum numbers referring to 
many-particle systems. 

F total angular momentum, including the contribution 
of nuclear spin. Interpretation as for/. 

/ nuclear spin quantum number, significance as for/. / 
may have integral or half-integral values, but one 
isotope of an element has a single, characteristic value 
of /.See Table 17. 

I J total angular momentum (excluding nuclear spin), or 
designation of a general angular momentum. J,J is 
never negative, and may be integral or half-integral 
depending on the system, 

magnitude of a.m.: hJ [/'(/' + 1)] 

number of projections on a specified axis: 2/ + 1 

magnitude of projection: see m. 
If/ is composed from/j and/ 2 , then permitted 
values are/ =/, + / a ,/, +/ 2 - 1, . . . |/, -/ 2 |. 

K component of a.m. about symmetry axis of an 

axially symmetric molecule: Kit K is restricted to 
the 2/ + 1 values, J.J-1, -J. 



S..L orbital a.m. quantum number (also called the 

azimuthal quantum number}. Interpretation as for 
/ but fi,L can take only integral values. 

m,M magnetic quantum number (often appearing as nip, 
m^ M L , M s , M f , Mj, etc.) 

component of a.m. on a particular axis (by 
convention the z-axis): mil, 
energy of a magnetic moment in a magnetic 
field B:-mhyB. 

n principal quantum number; the energy of an 
electron in the -hydrogen atom: —R In 2 with 
n = 1, 2, . . .. (Note that £ cannot exceed n — 1.) 

general quantum number in a variety of situations; 
for example, the -particle in a square well has 
n = 1 , 2, . . . and energy n 2 V/8mZ. 2 ). 

IV total angular momen turn, excluding electro n and 
nuclear spin. Restriction and interpretation as for 
U. 

S,S °spin a.m. quantum number. Significance and 

interpretation as for/V, and s,S may have positive 
half-integral values, s is a stngle-valuded intrinsic 
property of a particle. 

&$ spin projections for a spin-j object, a corresponds to 
m s - +~ and p 1 torn = -\. 

\A component of orbital a.m. about symmetry axis of a 
linear molecule: Ah, All A,Aare restricted to fi, £—1, 
. . ., — £ or L, L- 1, ,..,—£. respectively. \— Xare 
degenerate to a first approximation. 

V vibrational quantum number. The energy of a 
"harmonic oscillator is {JJ+ jlhoJo, where V ~ 0, 1, 
2 

0,2 component of spin a.m. about symmetry axis of a 
linear molecule: oft, 2ft a can lie ats, s— 1, . . ., — s, 
and 2 at S, 5-1 S. 

lj,£2 component of total electronic a.m. about symmetry 
axis of a linear molecule: ojfi, Slh. Q, can take the 
2/+ 1 va\uesJ,J-],...,-J. 



192 



quantum theory: a synopsis 



In Box 19 are listed some common quantum numbers and 
the properties they determine: for further information, consult 
the appropriate entry. The reader might care to reflect on why 
quantum numbers are always either integral or half-integral, 
and newer fractions more vulgar nor numbers irrational. 



quantum theory - , a synopsis. The view that energy could 
be transferred between systems only in discrete amounts rather 
than continuously arose from observations on the interaction 
of matter and radiation and on the behaviour of solids at low 
temperatures. The evidence came from the study of "black- 
body radiation, the "photoelectric effect, the -Compton effect, 
"atomic spectra (especially the spectrum of atomic "hydrogen), 
and the "heat capacities of solids. The first quantum calculation 
was due to Planck, who deduced the distribution law for 
black -body radiation. The first quantum-mechanical calculation, 
where the quantum ideas were applied to a mechanical system, 
was "Bohr's calculation of the energy levels of atomic hydrogen. 
The early theory of quantum mechanics was displaced by a 
new quantum theory in 1 926, when "Schrodinger proposed 
his equation, and Helsenberg his "matrix mechanics. These 
entailed a wholesale revision of classical physics, and gave a 
theoretical basis to the wave-particle -duality of matter. The 
incorporation of relativity into the theory was made by 
"Dirac, and relativistic quantum mechanics is now at the stage 
of 'quantum electrodynamics and quantum field theory. 

Further information .See MOM Chapter 1 for an outline of the 
observations that led to quantum theory. See also Heisenberg 
(1930) and Jammer (1966) for a historical perspective. An 
interesting introduction has been given by Andrade e Silva 
and Lochak (1969), A collection of the significant early 
papers (in translation) has been prepared by van der Waerden 
(1967). 

quenching. The angular momentum of a system is quenched 
when it is eliminated by the presence of some electrostatic 
potential. In an atom, and for simplicity we fix our attention 
on a hydrogen atom, the energy of an electron is independent 
of the angular coordinates (its latitude and longitude), and so 



its angular motion can occur smoothly and without hindrance. 
In such a case the "orbital angular momentum remains constant, 
and is well defined. When the atom is surrounded by ligands 
the energy of the electron depends on its angular coordinates 
and it experiences a force that accelerates it in a complicated 
manner. In classical terms the acceleration continuously changes 
the direction of the electron's motion and the average angular 
momentum is zero. 

One cannot use quite the same argument in quantum 
mechanics because the trajectory is an alien concept, but one 
can come to the same conclusion by considering the effect 
of the ligand potential on the wavef unction of the electron. 
The presence of the "ligand potential causes the electron to 
collect in pools of high probability, either close to the ligands 
if the potential is attractive or between them, if it is repulsive; 
but the formation of these pools implies that the original 
running wave has been turned into a standing wave {the 
stationary antinodes being the pools), and with standing 
waves there is associated no angular momentum. Therefore 
the momentum has been quenched by the anisotropic 
potential. 

Questions. 1 . What does 'quenching of angular momentum' 
mean? When does it occur? What causes it? What is its 
quantum-mechanical explanation? Can the same explanation 
account for the fact that in diatomic molecules only the 
angular momentum about the internuclear axis is well 
defined? 

2. A wave running around thez-axis is described by the func- 
tion exp i/n0; the ^component of the angular momentum is 
found by calculating the "expectation value of the operator 
{h/i)d/30. Show that the z-component of the angular momen- 
tum for this state is mil. Now quench the momentum by 
replacing the running wave, which has an even distribution of 
probability, by the standing wave cos m<ji, which has accumu- 
lations of probability in the vicinity of = and f. What is 
the expectation value of the z-component of angular momen- 
tum for this wave? 

3. Using the -hermiticity of the operator % g , prove that its 
•expectation value is necessarily zero for real states. This is a 
formal demonstration that angular momentum is quenched in 
states represented by real wavef unctions. 



Further information . Angular-momentum quenching is 
described in more detail in MQM Chapter 6 and Question 3 
is answered on p.417. Quenching profoundly affects the 
-magnetic properties of materials, in particular those of 
transition-metal ions, for when the orbital motion is eliminated 



quenching 193 

the paramagnetism can be ascribed solely to the spin magnetic 
moments. These aspects of quenching are also described in 
Earnshaw (1968), Ballhausen (1962), Figgis (1966), Jorgensen 
(1971), and Griffith (1964). See also Davies (1967), and 
van Vleck (1932). 



R 



radial distribution function. The radial distribution function 
(r.d.f.) determines the probability that a particle will be found 
somewhere within a spherical shell of thickness dr at the radius 
r. For a wavefunction depending on only the radius the radial 
distribution function Pir) is 4m- 2 ^*{r)^{r), and the probability 
of being in the shell of radius r and thickness dr is P(/)dr, The 
source of this function can be traced by recalling the inter- 
pretation of the -wavefunction and considering the response 
of a probe sensitive to the presence of the particle. The prob- 
ability of finding a particle in the volume element dr surround- 
ing the point r is equal to ^*{r)^(r)dr, and a probe of volume 
dr gives a response proportional to i^*(r)i^{r). Now let the 
sensitive part of the probe consist of a thin spherical shell of 
thickness dr and radius r. This probe is dipped into the atom 
so that the nucleus is at the origin of the shell, and the meter 
reading is indicative of the total probability of finding the 
particle anywhere on the shell. The volume of the shell is 
4ro- 2 dr, and so if the wavefunction is isotropic, the reading is 
proportional to 47rr 2 i/'*(r)i/'(r)dr, or P(r)dr. This means that 
P(r) tells us the probability of finding the particle anywhere on 
the shell of thickness dr at r. 

The probe behaves in an odd manner, because its sensitive 
volume diminishes as tt samples regions closer to the nucleus 
(the volume of the shell is proportional tor 2 ). The meter 
reading falls to zero at the nucleus, because there the shell 
becomes vanishingly small, and it also falls to zero at very great 
distances, because there the wavefunction vanishes. The 
decaying exponential wavefunction of the hydrogen atom 
therefore gives rise to a r.d.f. that rises from zero at the nucleus, 
passes through a maximum, and then falls to zero at infinity. 



This curve (see Fig. R1) shows the probability of finding the 
electron at a particular radius irrespective of the angular 
coordinates of the point: the square of the wavefunction itself 
gives the probability of finding the electron at a specified 
point, and the number of these points at a given radius increases 
as 4w 2 . 

Questions. What is the significance of the radial distribution 
function (r.d.f.)? What is the difference in interpretation of 
the r.d.f. and the square of the wavefunction? Discuss the 
form and significance of the r.d.f. for the electron in the 
1s-and2s-orbital5of the 'hydrogen atom. Using the math- 
ematical form of these two functions (Table 15) plot the 




Radius 

FIG. R1. The radial-distnbulian function for the ground state of 
hydrogen. 



194 



corresponding r.d.f.. Deduce an expression for the most 
probable radius of the charge distribution and discuss the 
nature of the r.d.f. for anisotropic orbitals (for example, 
the 2p-orbitals of hydrogen}. 

Further information, A discussion of the r.d.f. will be found 
in books that deal with atomic structure: see Chapter 2 of 
Coulson (1961), §1.4 of Herzberg (1944), §4.10 of White 
(1934), and §V.2 of Condon and Shortley (1963). Radial 
distribution functions enable one to think sensibly about a 
lot of inorganic chemistry, because they are the basis of 
•penetration and shielding and consequent discussions. For 
the trail through this matter 6ee Puddephatt's The periodic 
table of the elements (OCS 3) , pp. 34, 35 of Pass's tons in 
solution 3 (OCS 7), Earnshaw and Harrington's The 
chemistry of the transition elements {OCS 13), and Phillips and 
Williams (1965), particularly Chapters 1 and 2. Analytical 
expressions and references to the r.d.f. of numerous atoms 
will be found in McGlynn, Vanquickenborne, Kinoshita, and 
Carroll (1972), especially Appendix B. See also Herman and 
Skillman (1963) for numerical tables. 

Raman Spectra. The Raman process is the inelastic scattering 
of light by molecules. An inelastic process is one in which 
energy is transferred between the two colliding systems. In 
Raman scattering the light may deposit energy in the molecule 
by exciting one or more of its internal modes (of rotation or 
vibration), or it may collect energy from the molecules if a 
mode is already excited. Since the internal modes are 
"quantized the energy transfer is limited to well-defined 
amounts, and so the scattered light contains frequency 
components that are shifted from the incident frequency by 
discrete amounts. The detection and recording of the frequency 
composition of the scattered light constitutes the Raman 
spectrum of the species. 

In practice a brilliant, monochromatic beam irradiates the 
sample and the light scattered in the perpendicular direction is 
analysed: the forward -scattered component would be 
obliterated by the intense incident beam (but see below). The 
spectrum consists of a strong component at the incident 
frequency, the Rayieigh scattered component, which represents 
the elastic collision of the light with the sample, and a series of 



5-branch 



"I 



Raman spectra 



6B«- — « 



195 



5 

Stokes lines 



r 
2 

Rayieigh 
lire 



O-b ranch 



S 10 

anti-States lines 



incident frequency 
FIG. R2. Rotational Raman spectrum of a linear molecule. 

lines to high and low frequency (Fig. R2). The low-frequency 
lines are the Stokes lines, and arise from inelastic collisions in 
which energy is transferred from the light to the molecule. The 
lines on the high-frequency side of the Rayieigh component are 
the anti-Stokes lines, and arise from inelastic collisions in which 
molecular excitation is transferred from the molecule to the 
light. The intensity of the Stokes lines is greater than the 
intensity of the anti-Stokes lines because the latter depend on 
the presence of molecules already in higher-energy states. It 
should be clear that the spacing of the Raman lines contains 
information about the vibrational or rotational levels of a 
molecule, but to determine the information we must first decide 
the significance of the transitions. 

The Raman effect depends on the properties of the "polariz- 
ability of the molecule. This can be understood when the 
scattering process is pictured in terms of the incident radiation 
inducing a "dipole moment in the molecule, and this dipole 
moment radiating electromagnetic radiation. The efficiency of 
the process depends on the ease with which the molecule can 
be distorted by the incident light, and hence It depends on its 
polarizability. It can be understood on this picture that the 
emitted light will not necessarily carry away from the molecule 
all the energy needed to excite it: discrete amounts of energy 
can stick to the structure. The molecule, then, must be 
polarizable; but that alone is insufficient. 

If the molecule is to show a rotational Raman spectrum 
(where the inelasticity of the collision excites or deactivates 
rotational motion) its polarizability must depend on its orien- 



196 



Raman spectra 



tation, A rotating hydrogen molecule is Raman active because 
it has different polarizabilities parallel and perpendicular to 
the bond. The methane molecule is rotationally Raman inactive 
because, being spherical, its potarizability is independent of its 
orientation and the field cannot couple to the nuclear frame- 
work. In the case of the elastic, Rayleigh scattering, the 
requirement is much less stringent: all a molecule need be is 
polarizable; therefore all molecules scatter elastically, but onl. 
molecules with anisotropic polarizabilities can exchange 
rotational energy with the light. 

If the molecule is to show a vibrational Raman spectrum 
the polarizability must change as the molecule vibrates. A 
vibrating hydrogen molecule is Raman active because its 
polarizability depends on how greatly the bond is stretched; 
the antisymmetrical vibration of CO2 , which we can denote 
CH-«-C— *0 ^ CH — C-*«-0, does not affect the polarizability of 
the molecule (it leaves it roughly the same size) and so this 
particular vibration is Raman inactive. As in the rotational 
case, we see that the change in the polarizability is the 
essential feature if energy is to be exchanged and the collision 
be inelastic. 

An important rule, the exclusion rule, applies to the 
vibrational Raman spectrum of molecules having a centre of 
symmetry. If in such a molecule there is a mode of vibration 
active in the ordinary {infrared) "vibrational spectrum, then 
that mode is inactive in the vibrational Raman spectrum; 
conversely, if the mode is infrared-inactive, it is Raman-active. 
It follows that the Raman effect is useful in the study of 
vibrations (and rotations) that are inaccessible to normal 
absorption spectroscopy. 

The scattered radiation of the Raman effect is polarized 
even if the incident radiation is not. Fig. R3 illustrates the 
simpler case of Rayleigh scattering, which is the elastic 
scattering of light: the Raman scattering process is analogous, 
but slightly more involved. From Fig. R3 a it should he clear 
that if the molecule is isotropic and the incident light plane- 
polarized the scattered light is also plane-polarized. The 
scattered light is also plane-polarized if the incident light 
is unpolarized. If the molecule is anisotropic (Fig, R3 b), 
the scattered light has both polarization components, and 
so is not plane-polarized. A convenient measure of the degree 
of polarization is the depolarization ratio p, which is defined 





FIG. R3. Depolarization of Rayleigh lines and, by analogy, of Raman 
lines. 

as the ratio of the intensities / |( and ^as defined in Fig. R3. 
In the isotropic molecule case there is zero intensity in the 
scattering plane, and so p = 0. In the anisotropic case there 
is intensity both in the plane and perpendicular to it, and 
the depolarization ratio is non-zero. For a freely rotating 
molecule the maximum value is p = 7 for unpolarized 
incident light, and p= | for plane-polarized incident light 
{for the polarization and geometry of Fig. R3). The 
polarization in Raman scattering is determined similarly, 
but it is necessary to consider the isotropy or anisotropy of 
the changes in the polarizability of the molecule. Thus a 
completely symmetrical vibration plays in the Raman case the 
role of the symmetric molecule in the Rayleigh case; therefore 
the Raman scattering from such a mode is fully polarized and 
p = 0. If a vibration is anisotropic the Raman scattered light 
is depolarized, and so p > 0. If the incident light is polarized 
p cannot exceed |, and if it is unpolarized p cannot exceed 
7-. It follows that the determination of the polarization of the 
Raman scattered light is a valuable tool for determining the 
symmetry of the active molecular vibrations. 

We still do not know what transitions the Raman lines 
represent apart from the qualitative remark that they represent 



modes involving a changing polarizability. The 'selection rules 
for Raman-active transitions are that the "vibrational quantum 
number of a mode must change by + 1; the Totationat quantum 
number must change by AJ = + 2, The two-quantum rotational- 
Jump behaviour stems from the fact that the transitions depend on 
the polarizability rather than the 'transition dipole moment 
(which determines the absorption spectrum). One approach 
to understanding the occurrence of 2 is to picture the scattering 
process as involving two dipole transitions, one for the photon 
coming in and one for the photon going out. A classical picture 
elucidates why the rotational quantum number changes by 
± 2 but the vibrational quantum number changes by ± 1 : in a 
rotation of a molecule the dipoie moment is restored to an 
indistinguishable orientation after a complete rotation of the 
molecule, but the polarizability is restored to an indistinguish- 
able orientation twice on a revolution. In a vibration the dipole 
moment and polarizability return at the same time and 
vibrational Raman and absorption spectra both have Av = ± 1. 

The rotational Stokes and anti-Stokes lines are related 
to the "branches of a rotation-vibration spectrum; and anal- 
ogously to the notation used there, they are referred to as the 
O-branch (AJ = -2; anti-Stokes) and the S-branch {AJ m +2; 
Stokes). Knowing what transitions the spectrum shows it is 
possible to relate the Raman lines to the energy levels of the 
molecule by the same analysis as in the corresponding absorption 
spectrum, and so to extract "force -constants, moments of 
inertia, and molecular geometry, and to identify unknown 
species. 

An important experimental requirement in Raman 
spectroscopy is an intense monochromatic light beam; what 
is more natural than to apply -lasers, which have just this 
property? Laser-Raman spectroscopy is today a major branch 
of study; not only does it refine the conventional Raman 
technique by enabling frequencies very close to the exactly- 
defined incident frequency to be studied, but its unique 
properties give rise to a number of new effects. The properties 
put to work are the low divergence of the beam, which enables 
observations to be made close to the forward direction, and 
its exceedingly high intensity. 

In the stimulated Raman effect an intense laser beam is 
focused on a sample, and the light scattered in the forward 
direction, or just off -axis, is observed. The light in the forward 



Raman spectra 



197 



direction itself is the same as the incident frequency, but 
mixed with it is Stokes radiation of frequencies v,v— v., 
V— 2l> { etc., where v, is the frequency of an internal mode 
giving an intense Raman signal in a conventional experiment. 
Surrounding this narrow, forward- scattered beam are a series 
of concentric circles of light of increasing frequency; the 
first ring is of frequency v + v the next ring has frequency 
v+ IV y the third P+3i>, and so on (Fig. R4 a). The effect 
arises from the fact that the initial Stokes scattered line is so 
intense that it can be scattered again, and an inelastic collision 
leads to a further Stokes line at V — 2v.; and so the process 
continues as more quanta are chipped off the beam. Likewise 
the intensity of the anti-Stokes lines is due to successive scat- 
tering, and the initial high intensity of the anti-Stokes light at 
u+y. f is due to the high intensity of the initial beam permitting 
the annihilation of two V photons by one molecule, followed 
by the simultaneous generation of one of frequency V + V. and 
another of frequency V — V y Only for very high beam intensities 
is there a significant probability that two photons can simul- 
taneously be in the region of the same molecule, which is 
necessary if the frequency-sharing is to occur. The angular 




FIG. R4. (aj Stimulated Raman effect, (b! Hyper-Raman effect. 
Ic) Inverse Raman effect. 



198 



Ramsauer effect 



dependence of the frequency of the scattered light is a conse- 
quence of the conservation of momentum in the collision. 

In the hyper-Raman effect an intense beam of frequency P 
is focused on a sample, and together with the normal Raman 
scattering there appear frequencies of 2i>and 2v± t>, (Fig. R4 b). 
Thus the hyper-Raman effect is the inelastic scattering struc- 
ture on high-frequency photons generated by the annihilation 
of two low-frequency photons in a simultaneous event involv- 
ing one molecule and two incident photons. As in the stimu- 
lated Raman effect, the efficiency of the hyper-Raman effect 
depends on the intensity of the light. One application is to the 
measurement of molecular hyperpolarizabitities (see 
"polarizability). 

In the inverse Reman effect two beams of light are employed 
and focused on a sample. One has a continuous spectrum and the 
other is a highly intense monochromatic beam. It is found that 
transitions occur which appear as absorptions from the 
continuum at frequencies V + v . The process may be pictured 
as the arrival at the molecule of a photon of the continuum 
with a frequency V+ V; simultaneously there arrives a laser 
photon of frequency V which stimulates the excited molecule to 
shake off a photon of its own frequency, leaving behind an 
amount of energy corresponding to the frequency V. (Fig. R4 c). 

Questions. 1 . What is the difference between an elastic and an 
inelastic process? On which does the Raman effect rely? What 
experimental arrangement is employed in Raman spectroscopy? 
In what way can the use of a laser benefit the observations? 
What is meant by the Rayleigh component? Discuss the 
appearance of the Raman spectrum and indicate the signifi- 
cance of the Stokes and anti -Stokes lines. On what molecular 
property do the Raman and Rayleigh scattering rely? Why do 
they so depend? What is a necessary property of the molecule if 
it is to show a vibrational Raman spectrum, and a rotational 
Raman spectrum? State the exclusion rule. Which vibrations of 
carbon dioxide are Raman -active, and which are infrared-active? 
State the selection rules for vibrational and rotational Raman 
spectra: why does the latter reiy on a double-quantum jump, 
but the former on a single jump? What information can be 
obtained from the splittings in the 0- and S-branches of a 
rotational Raman spectrum? 
2. Show that in a classical polarizable rotator the induced 



dipole moment will emit Stokes and anti -Stokes lines. In the 
calculation suppose that the polarizability of the molecule 
varies harmonically between a — 6a and a + 5a about a mean 
value a. Proceed by showing that the molecule possesses an 
induced dipole moment of the form aE + oofcostor, where 
E is the imposed optical field; then take the time-dependence 
of £ to be costdof, and show by simple trigonometric manipu- 
lations that the overall induced dipole has components, 
oscillating at OJ, oj± av This calculation leads to the prediction 
of equal intensities for the Stokes and anti-Stokes lines: why 
is that false? 

3. "Group-theoretical arguments may be applied to the Raman 
problem in order to determine the vibrational selection rules. 
The transition operator is the polarizability tensor for the 
molecule: this transforms likexx, yy, and zz. Show, using 
Box 4 on p. 33, that in a centrosymmetric molecule the exclusion 
rule follows from the difference in symmetry of the polariz- 
ability and the dipole moment under inversion. From the 
characters in Table 3 confirm that the A i( E, and T 2 vibrations 
are Raman active in a tetrahedral AB 4 molecule. What trans- 
itions are Raman active in water, carbon dioxide, and 
ammonia? 

Further information. See MQM Chapter 10 for answers to 
Questions 2 and 3 and for further discussion. For applications 
see Woodward (1972), Wheatley (1968), Whiff en (1972), 
and Barrow (1962). Laser-Raman spectroscopy is described in 
Long (1971) and Gilson and Hendra (1970). See Herzberg 
1 1945) for further details. Woodward's Chapter 19 is a good 
summary of depolarization processes. 

Ramsauer effect. When a beam of electrons was passed 
through a sample of argon and the other noble gases, it was 
found that the scattering power of the sample decreased 
strongly at some energies of the electron beam (a transmission 
resonance is observed at about 0.7 eV). This effect may be 
understood in terms of the 'wave nature of the electron, and 
its decreasing wavelength as its energy increases { °de Broglie 
relation). The system may be viewed as an electron wave 
incident on atoms, and these are regions of potential different 
from the surrounding vacuum. But in regions of different 
potential energy the wavelength of the electron is changed 



refractive index 



199 



(see -kinetic energy). Just as in the case of an incident beam 
of light passing into a region where the refractive index 
changes, there are reflections from the front of the atom 
(where the potential changes abruptly), and reflections from 
the opposite inside surface where the potential drops back to 
the vacuum level. If the potential has a thickness equal to 
one quarter of the wavelength of the electrons the waves 
reflected backwards from the two surfaces interfere destruc- 
tively, but the waves transmitted interfere constructively. It 
follows that the intensity of the electrons reflected by the 
atom is reduced and that the transmitted intensity is increased. 
(This of course is the role of a coating on a lens.) 

Further information . A helpful discussion of transmission 
resonances is given in §11. 8 of Bohm {1951} and applied to 
the Ramsauer effect in §11. 9 and §21.54, where he explains 
that the effect is not quite the same as the square- well process 
we have described. See also §99 of Davydov (1965) and Mott 
and Massey (1965). 

refractive index. The ratio of the speed of light in vacuo c to 
its speed in a medium v is the refractive index of the medium: 
n = civ. 

The size of the refractive index depends on the strength of 
the interaction between the light field and the medium, and as 
the electric field of the light has a stronger interaction than 
the magnetic field, we should expect the refractive index to 
depend on the electric "polarizability of the medium. The 
greater the polarizability, the stronger the interaction, and the 
greater the drag on the progress of the light. This guess is 
confirmed by calculation, for in a nonpolar medium the 
refractive index is related to the relative permittivity (dielectric 
constant) by ri 1 = € r , and the relative permittivity Is related 
to the polarizability by e r = 1 + Nct(Uile , where N is the 
concentration of molecules and a(co)their polarizability. But 
we have to be just a little careful because the molecule on 
whose polarizability we might at one instant focus our 
attention is surrounded by other polarizable molecules. These 
other molecules respond to the electric field of the light and 
their polarization enhances the field experienced by the 
central molecule. Therefore we should apply a correction to 
take into account the presence of the surrounding molecules. 



This is the Lorentz local -field correction, and it involves 
increasing the strength of the field by a factor of | (e +2), 
e is the relative permittivity, or n , and so the expression 
for the refractive index becomes rather more complicated. 
By an unlikely but helpful coincidence the indistinguishable 
Lorenz and Lorentz introduced this correction independently 
and simultaneously, and arrived at the Lorenz- Lorentz formula 
(n 2 - 1 )l{n 2 + 2)p = Afa:M/3pe o , where p is the density. 
Since N is proportional to the density the right-hand side of 
the equation is independent of the density, and so too 
therefore is the term on the left, which is called the refractivity 
of the medium. 

It should be noticed that the refractive index (and the 
refractivity) depends on the frequency of the light through 
the dependence of the polarizability on the frequency. This 
dependence is described in the section on "polarizability, but 
at optical frequencies is due to the high-energy photons being 
more able than low-energy photons to excite the molecules 
into their low-lying excited electronic states. Therefore, as 
the frequency gets greater (and approaches an absorption 
frequency of the molecule), the interaction gets stronger and 
the refractive index gets larger. For this reason the refractive 
index for blue light exceeds that for red light, and, as a 
consequence of this, a beam of white light is dispersed by a 
refracting medium. The name dispersion, which denotes the 
frequency-dependence of a property, is derived from this 
aspect of the refractive index. 

Very close to absorption bands the refractive index varies 
strongly. If the behaviour of the refractive index throughout 
the frequency range is known it is possible to extract the 
absorption spectrum of the molecule and vice versa. The 
formula that enables this to be done is the Kramers- Kronig 
dispersion relation. A dispersion relation relates the overall 
frequency-dependence of a dispersion property (such as 
refractive index) to the absorption property and vice versa. 

The refractive index for a composite molecule is the sum of 
the refractivities of its parts in so far as the polarizability of a 
molecule is itself an additive property. 

Further information. See MQM Chapter 1 1 for an account in 
more detail of the derivation of the quantum-mechanical 
expression for the refractive index of the molecule. See 



200 



Renner effect 



Corson and Lorrain (1970) for a derivation via the Clausius- 
Mossotti equation of the Lorentz local-field correction. See 
also van Vleck (1932) for a discussion. A simple introduction 
to the Kramers-Krdnig dispersion relation will be found in 
Slichter (1963), and tougher accounts in Roman (1965, 1969). 
Pertinent information on refractive indices is contained in the 
polarizabitity Box (Box 17 on p. 179), 

Renner effect. The Renner effect, or as it is sometimes more 
fairly called the Renner-Telter effect, is an interaction between 
the electronic and vibrational motions of a linear molecule 
(especially a triatomic linear molecule) which removes the 
"degeneracy of the energy levels. Consider a II-state of a linear 
triatomic; bending the molecule into a triangular conformation 
affects the two components of the electronic molecular 
orbital differently. For simplicity let the molecule have a 
single electron in a jt -orbital; in the linear case the it x -and 
it -orbitals are degenerate, but when the molecule is bent 
they diverge in energy: three possible types of behaviour are 
illustrated in Fig. R5. The Renner effect appears in the 
spectrum of the molecule because the bending vibrational 
levels are modified by their interaction with the electronic 
levels. 



been given by Renner (1934), Pople and Longuet-Higgins 
(1958), and Longuet-Higgins (1961). 

resonance. The concept of resonance has its roots in classical 
mechanics, and it is helpful to recall that application of the 
term. If two pendulums are weakly linked (for example, if 
they hang from the same slightly flexible axle, as in Fig. R6) 
the motion of one is experienced by the other. If one is 
initially still, and the other set in motion, the energy of the 
latter will be transferred to the first, which will begin to 
swing, and pass back to the second some of its acquired energy. 
This ebb and flow of energy continues indefinitely in the 
absence of damping forces. The exchange of energy is most 
effective when the pendulums have the same natural frequency, 
and this condition of equality is known as resonance; the 
energy, or amplitude, is then said to resonate between the two 
tuned systems. 

It is possible to imagine a form of coupling where the energy 
of the system is lower if the pendulums swing in phase, and 
where the worst arrangement (in terms of energy) occurs when 
they swing in opposition. A flexible axle is an example: it 
might require less energy to twist the support in the same 
direction at each pivot than to twist it in opposition (Fig R6); 




O^X) 





FIG. R5. Renner effed splitting 
of ir-levels: three possible 
situations. 



Further information . An account of the Renner effect in 
molecules is given in §1.2 of Herzberg (1966), who provides 
a number of examples, and gives expressions for the energies 
of the vibronic states of the molecules showing the effect. See 
also §10.13 of King (1964). A quantitative treatment has 



if there is no coupling between the periodic systems their 
relative phase is immaterial. 

With the preceding classical picture of resonance and 
coupled systems in mind we shall consider two examples of 
quantum-mechanical resonance. The first is the interaction of 




FIG. R6, Resonance of two coupled pendulums. The energy splitting A 
is greatest when the natural frequencies are identical. 

light with atoms (or molecules) . The atom and the electro- 
magnetic field (light) play the roles of the two coupled 
periodic systems. The 'natural frequencies' of the atom are 
the transition frequencies, and we can imagine adjusting the 
'natural frequency' of the electromagnetic field bathing the 
atom by changing the frequency of the incident light. There 
comes a point when the frequency of the light matches the 
frequency of a transition within the atom: the combined 
system behaves like the two coupled pendulums, and energy 
is transferred from the light field (resulting in absorption) or 
to the light field if the atom is already in an excited state 
(emission). These processes occur most effectively at resonance. 

The other example of resonance is the type one encounters 
tn theories of molecular structure, particularly in the "valence- 
bond theory, where one attempts to describe the true structure 
of a molecule by a superposition of simple, canonical structures. 
The best-known example is -benzene, where one attempts to 
describe the structure by the superposition of the two Kekule 
structures. Let us suppose that there is a coupling between the 
Kekule forms; then the superposition will have a lower energy 
than either form alone if the phase of coupling is correct 
(remember the second aspect of the classical idea of resonance). 



resonance energy 201 

Thus each Kekule' structure behaves like a simple pendulum (by 
symmetry they correspond to the same frequency) and the 
presence of a coupling means that their energy is different in 
conjunction than separately. The nature of the coupling may 
be visualized as the tendency of the n -electrons in one C— C 
bond to push the neighbouring ir-electrons into a neighbouring 
gap in the jr-bonding structure: in this way one Kekule structure 
is turned into the other (Fig. R7) (see "benzene and Fig. B8). 
Thus resonance stabilizes in the sense that the coupled system 
has a lower energy; the stabilization is greatest when the natural 
frequencies of the separate systems are the same (consequently 
the resonance stabilization of benzene is large), lonic-covalent 
resonance also stabilizes, but as the contributing forms differ 
in energy the resonance is less exact and the effect less. 



our of phase 




dec trostofic 

coupling 



in phase 



FIG. R7. Resonance in VB theory (of benzene). E denotes ihe 
resonance energy. 

Further information. For discussions of the concept of resonance 
in molecular-structure studies, see §5.5 of Coulson (1961), §1.3 
of Pauling (1960), and §13 of Eyring, Walter, and Kimball 
(1944). Resonance in light absorption is discussed in Chapter 7 
of MQM, and very well illustrated by the phenomena of 
"electron spin resonance and "nuclear magnetic resonance. A 
transmission resonance is described under "Ramsauer effect. 
See also "benzene, "valence bond, and "resonance energy. 

resonance energy. An "aromatic molecule is more stable than 
untutored speculation, confronted with a molecule bristling with 
double bonds, might predict. The difference between the true 
energy, which takes into account the stabilization of the 
molecule by "resonance among various structural possibilities, 



202 



rotation of molecules 



and some reference state of the molecule, is termed the 
resonance energy (see Fig. R7). {An alternative name, which 
reflects the analogous phenomenon in 'molecular-orbital 
theory, is derealization energy.) 

The choice of reference is difficult and several suggestions 
have been made. The Huckel definition is the most elementary: 
it defines the resonance energy as the difference in energy 
between the true molecule and the hypothetical molecule in 
which there are localized ethene-type double bonds. Thus if 
the jr-electron energy of 'benzene is found to be 3a + 40, and 
the energy of each of the three ethene bonds is a + j3, the 
resonance energy is |3. 

The difficulty with the Hu'ckel definition is the arbitrariness 
of the comparison: would it not be better to attempt to find 
the energy of the unconjugated form of the molecule itself, and 
then use this as the reference? Why not take the hypothetical 
cyclohexatriene molecule, calculate its energy, and use it as 
the reference for benzene? This reference molecule has 
alternating short and long bonds, and it is argued that by its 
use the full effect of conjugation, derealization, or resonance 
on both the 7T- and the a-electrons is taken into account. This 
is the basis of the Multiken-Parr definition of resonance energy. 

A common method of measuring the resonance energy is to 
determine the heat of formation of the molecule (for example, 
by determining the heat of combustion of the molecule in a 
bomb calorimeter, and knowing the heats of formation of 
the combustion products, which are often carbon dioxide 
and water), and comparing this result with the value obtained 
on the basis of ascribing a bond energy to each bond in the 
structure (these bond energies may be found from tables): the 
difference of the two calculations is the resonance energy. For 
example, in benzene a Kekule structure has six C— H bonds, three 
C— C bonds, and three C=C bonds; the energy of that structure 
is therefore 6F(C-H) + 3F(C-C) + 3£(C=C). The difference 
of this value from the observed heat of formation is the 
resonance stabilization of the Kekule' structure (150 kj mof 1 )- 
Modern values are often obtained from the heat of hydrogen- 
ation, which is less in the presence of resonance stabilization 
(the more the molecule is stabilized the closer it lies to the 
fully hydrogenated energy). In this type of determination a 
molecule containing three (unconjugated) double bonds is 
expected to have a heat of hydrogenation three times that of 



cyclohexene;the observed heat of hydrogenation of benzene 
is less than that figure by 1 50 kJ mol -1 , and this difference is 
identified as the resonance stabilization of the Kekule form. 

Questions. What is meant by resonance energy? What are two 
possible definitions? Can you think of alternative definitions? 
How would you determine the resonance energy of a hydro- 
carbon? What changes in the value of the resonance energy 
might you expect on replacing the Hu'ckel definition by the 
Mulliken-Parr definition? The heat of formation of the 
naphthalene molecule was measured and found to be 8623 kJ 
mol -1 . The following bond energies have been measured in 
other experiments: C-C, 333 kJ mol" 1 ; OC, 593 kJ mof 1 ; 
C-H, 418 kJ mof' . Calculate the resonance energy of 
naphthalene. On the basis of the Hu'ckel molecular-orbital 
scheme, estimate the resonance energies of cyclobutadiene and 
butadiene {buta-1, 3-diene). 

Further information. See Streitweiser (1961), Salem (1966), 
Dewar (1969), Murrell and Harget (1972), and McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972) for further 
discussion. A short table of resonance energies will be found 
in Chapter 9 of Streitweiser {1961 1 and Chapter 4 of Murrell 
and Harget (1972). A book devoted to resonance in organic 
chemistry is that by Wheland (1955|. 



rotation of molecules. The rotational energy of a molecule 
arises from its 'angular momentum, and because the latter is 
quantized, so too is the rotational energy. The energy separ- 
ation of adjacent quantized levels is small, and transitions 
between them occur in the microwave region of the spectrum 
(10 8 - 10 n Hz, 10 1 - 1CT 2 cm). The separations are deter- 
mined by the moments of inertia of the molecule, and so 
microwave spectroscopy gives information about molecular 
geometry. Rotational transitions are also observed in conjunc- 
tion with the "vibrational, 'electronic, and "Raman spectra of 
molecules, and these give the same kind of information. 
Molecules are normally classified into four groups In a dis- 
cussion of their rotational energy levels: linear molecules, 
symmetric tops (molecules with an axis of symmetry), 
spherical tops (spherical, tetrahedral, and octahedral molecules), 
and asymmetric tops {anything else). 



rotation of molecules 



203 



1. Linear molecules. A linear molecule has only two rotational 
degrees of freedom: these correspond to end-over-end rotation 
about two perpendicular axes. It is easy to see why the question 
of a third degree of rotational freedom does not arise: in a 
diatomic molecule the atoms have six degrees of freedom; 
three are ascribed to the overall translation of the molecule- 
one is ascribed to the vibration of the bond; and so only two 
remain for the rotations. A similar argument applies to any 
linear molecule. The classical kinetic energy of rotation of a 
body of moment of inertia / is jlco 2 , and as the classical angular 
momentum is /tothis energy may be expressed as {/Gj} 2 /2/. 
The transition to quantum mechanics is now trivial, for the 
quantum theory of -angular momentum tells us that the 
angular momentum of a body is limited to the values 
[J{J + 1 )] '^h with J confined to the integers 0, 1 , 2 There- 
fore it is merely necessary to replace /toby this expression to 
obtain the energy of the Jth quantized rotational level as 
jy + 1 )h 2 /2/. The quantity h 2 /2/ is normally written B and 
called the rotational constant oi the molecule. It is important to 
note that the separation of the rotational levels decreases as 
the moment of inertia of the molecule increases (for B then 
decreases} and that the level J lies at an energy 2BJ beneath 
Its neighbour. Typical values of B are 60.809 cm" 1 for Hj, 
30.429 cm" 1 for D 2 , 10.5909 cm" 1 for HCI, and 0.0374 cm" 1 
for l 2 ; more values are given in Table 10. 

In accord with the theory of -angular momentum there are 
2J + 1 possibilities for the orientation of the rotational angular 
momentum vector of the molecule, but in the absence of 
external fields the orientation of the molecule has no effect on 
its energy, and so 2J + 1 orientations all have the same energy 
(they are "degenerate). 

Pure rotational transitions can be stimulated by an electro- 
magnetic field of the appropriate frequency only if the 
molecule has a permanent dipole moment, for the dipole acts 
as a kind of lever for the interaction, and through it the field 
accelerates the molecule by exerting a torque. Molecules 
without dipole moments (including, for example, H 2 and C0 2 ) 
do not show a pure rotational spectrum. If a molecule has a 
permanent dipole the field can induce transitions only between 
neighbouring levels; that is, the "selection rule for rotational 
transitions is &/ = +1 (absorption) or A./ = — 1 (emission). The 
rotational spectrum is therefore a set of fines spaced by 2B, 



J: 6- 



J ^=L 



1 2 3 4 5 6 

FIG. R8. Pure rotation spectrum of a linear molecule, showing 
population of levels and the intensity distribution (which is determined 
by the population and the size ol transition dipole moments). 



with an intensity distribution governed by their initial (thermal) 
population and, in a slightly complicated way, by selection 
rules which vary with J (Fig. R8). 

2. Symmetric tops. Symmetric tops may be either prolate 
[cigar shaped) or oblate (disc shaped), and have three rotational 
degrees of freedom. As they have two distinct moments of 
inertia (one parallel to the figure axis /, , , and two equivalent 
moments perpendicular to the axis L, the energy of rotation of 
the molecule depends on how the angular momentum is 
distributed about the three molecular axes. If the molecule 
rotates end-over-end the energy is determined by /j_alone, but 
as the motion becomes more like the spinning of a top about 
its axis, so the energy becomes dependent more strongly on I... 
The amount of rotation about the figure axis is determined by 
a quantum number K: Kh is the component of the angular 



204 



rotation of molecules 




FIG. R9. Rotation of a (prolate) symmetric top. When K is large the 
motion is largely about the figure axis; when K is small it is largely 
about the perpend icu tar axis. 



momentum on the axis, and can vary in integral steps from 
+J to —J (there are 2 J + 1 such values) (see Fig. R9). When 
K = there is no rotation about the axis. It is quite easy to 
see how the quantum number K enters the problem: the 
classical energy of rotation depends on the angular momentum 
3bout each of the axes, If with the axis q there is associated a 
component of angular momentum Uoj) and a moment of 
inertia / then the kinetic energy associated with this mode is 
~{taj) 2 ff . Therefore the total kineticenergy of our molecule 

is J (/to}^// L + \ V<4 2 Jlj_+ \ U<4j// n , where z is taken to 
be the figure axis. If we recognize that \I<A\ + (/to)^ + {tco) 2 , 
is the square of the total angular momentum of the molecule, 
which can be identified with the quantum-mechanical value 
J{j + 1)fi 2 , it is possible to cast the energy expression into the 
form JU + \\B + j(1//„ - 1//J {100)1, where B is h2/2/ _i_- 

Next we identify (l(+>) z as the component of the angular 
momentum on the figure axis, and from our knowledge of 
the quantum theory of -angular momentum recall that all 
components of angular momentum are quantized, and so set 



this component equal to some integral multiple of h: thus we 
identify {/w) r with KTi. 

The final form of the energy expression is given in Box 20. 
The form of the expression is as we anticipated, for when K 
is zero the energy is determined solely by (j_, and when K - ±J 
the energy is determined largely (but not solely) by r"„. It is 
also satisfying to note that the energy depends on K 2 rather 
than K itself: this means, as common sense requires, that the 
direction of motion about the figure axis is immaterial. Remem- 
ber that the molecule can still have 2J + 1 different orien- 
tations with respect to some space-fixed axis, but these are 
-degenerate in the absence of external fields. 

The spectra of symmetric tops depend on the presence of 
a permanent dipole to act as a lever: by symmetry this dipole 
must lie along the axis and so a light wave is unable to exert 
a torque to accelerate the motion about the axis; consequently 
the "selection rules are AK = and AJ = ±1. 

3. Spherical tops. The energy levels of spherical tops can be 
obtained very easily from the expression for the symmetric 
top because we may set /j_= /,, . The resulting expression is 
given in Box 20. The energy is independent of K because all 
axes of the molecule are equivalent. Another way of saying 
this is that all the 2J + 1 states K for a given value of J are 
■degenerate. Therefore the rotational levels of the spherical 
top are [2J + 1 ) 2 -fold degenerate (because the 2J + 1 space 
orientations, labelled M, are also degenerate). A spherical 
top is invisible in microwave spectroscopy because it has no 
permanent dipole, and so cannot be accelerated by incident 
radiation. The rotational levels of this and the other types 
of molecule can, of course, be excited by collisions with 
other molecules, or the walls of a vessel, and so the large 
number of accessible rotational levels must be taken into 
account when the properties of the gas (such as its -heat 
capacity) are calculated. 

4, Asymmetric tops. These are real horrors because the angular 
momentum cannot be distributed among the three axes in a 
tidy fashion. If we think of a prolate symmetric top with a 
bump added to one side we shall see the problem. If the top is 
spinning about its figure axis the effect of the bump will be to 
tip the molecule away from its original direction; that is, the 
bump induces the transfer of angular momentum from the 
figure axis to the other two axes. But the process continues. 



rotation of molecules 



205 



BOX 20; Rotational energy levels 
Diatomic (and linear} molecules 

E{J)=BJ(J+%) B = tf/2t. 
(B is in joules if / is in kg m 2 ; B = h/400ffi:/ is the 
appropriate expression in cm" 1 .) 

Transition energy: A£(J) = E (J + 1) - E{J) = 2SI/ + 1 ), 
Symmetric tops 

EU, K\ = BJU + 1) + (A - B)K 2 
B = h 2 /2/ ± A=h 1 /2I U . 
Transition energy: Af (J) = £(J + \,K)— EU, K) 

= 2B{J + 1). 
Spherical tops 

E(J) = BJ{J+\) B = \\ 2 I2I. 
Moments of inertia: 

diatomic molecule: {m ,m B separated by R) 



J m A m B \ , 



linear triatornic molecule: (m m , m separated by 

fl A8 and V 

l = m tf +m R 2 -to^M-m^R ) 2 

m A rt AB^ m c W BC (BI A +/» a +m c ) 

pyramidal molecule: (AB 3 , bond length R, BAB 
angle 0) 

t n = 2m B R 2 i-\-co$0) 

« m .m B R 2 (1 + 2 cos&) 
J- B (3m. +m D ) 



and although the total angular momentum remains constant, 
the molecule rotates in a complicated, varying pattern. The 
energy levels can be obtained only by complicated techniques, 
and the rotational spectra are immensely complex. Neverthe- 
less, information about molecular geometry can be extracted 
by a close analysis of the shapes of the spectral bands. 



Questions. 1 . What is the source of the rotational energy of a 
molecule? Why is it quantized? Is there a zero-point energy? In 
what region of the spectrum do rotational transitions occur? 
How do the transition energies depend on the size of the 
molecule? What information can be obtained from rotational 
spectra? What must a molecule possess if it is to absorb 
radiation by a rotational transition? What is a classification of 
molecular types? What is the significance of the quantum 
number K? When does the rotational energy of a molecule 
depend on the space projection Ml How many-fold 
degenerate are the rotational states of linear molecules, and the 
various kinds of tops? Why does the energy of a symmetric top 
still depend weakly on /^when K = ±J? Can the rotational 
energy of a symmetric top ever be ascribed solely to motion 
about its figure axis? What are the selection rules for rotational 
transitions? How do all the rotational modes come into thermal 
equilibrium even though some transitions are electric-dipole 
forbidden? 

2. Calculate the energy-level separation (in cm -1 ) for the ro- 
tational levels of H 2 , HD, and D 2 . taking the bond length to be 
0074 nm in each case. The pure rotational spectrum of HI 
consists of a series of equally spaced lines with a separation 

of 12.8 cm -1 ; calculate the bond length of the molecule. Why 
are the lines equally spaced? {Draw an energy -level scheme 
and apply the appropriate "selection rules; or just think about 
2BJ and Ay.) 

3. Calculate the relative populations of the rotational levels 
of H 2 ,D 2 , l 2 (5 = 0.0374cm" l ),CH<, (r CH =0.1094 nm), 
and NH 3 (/,, = 4-437 X 10 -47 kg m 2 , l ± = 281 6 X lO" 47 
kg m 2 ) at 300 K and at 1000 K. Use the Boltzmann distri- 
bution, include the degeneracy of the levels properly, and 
forget (if you already know) about "nuclear statistics. 

Further in formation . A fuller discussion of molecular rotation 
and rotational spectra is given in MQM Chapter 10. See also 
Whiffen (1972), Barrow (1962), and King (1964). An 
introduction to microwave spectroscopy has been written by 
Sugden and Kenney (1965), and a standard work is that by 
Townes and Schawlow (1955); both books give a bibliography 
of molecules studied. Allen and Cross (1963) give a theoretical 
discussion. See Herzberg (1945, 1950, 1966) for further 
discussion and applications; in these volumes are useful collec- 



206 



Rydberg constant 



tions of molecular data (such as bond lengths, "force- constants, 
and rotational constants). A useful collection of molecular 
structural data has been compiled by Sutton (1958). Rotational 
transitions can also be studied by observing the "vibrational and 
"electronic spectra of molecules, and by thG use of "Raman 
spectra. The "heat capacity of molecules depends upon the 
accessibility of rotational states, and therefore on their energy. 
A complicating feature of rotational problems is that the 
occupation of rotational states is restricted by the "Pauli 
principle: for more information see "nuclear statistics. 

Rydberg constant. The Rydberg constant relates the energy 
of an electron in a -hydrogen atom to its principal "quantum 
numbers: E = —Rlri 1 . It is necessary to be just a little careful 
in recording values of R because it depends on the mass of the 
electron and the proton, and if some one-electron atom other 
than hydrogen itself is being considered it is necessary to 
correct it for the mass of the new nucleus. That having been 



BOX 21: 


Rydberg constant 


Rydberg constant 

rV = &m a e*c 3 m 3 

R' m C 


1-097 373 X 10 7 m" 1 
1-097 373 X 10 s cm"' 
3-289 842 X 10 1S Hz 


fl „ = R l hc = 2™ e c2a 


'■ 2-179 72 X 10~ 18 J 

1313kJmol"' 
13-60 eV, 


Hydrogen-atom Rydberg constant 

r' = R'l(\ + m Im ) 1-096 776 X 10 7 m" 1 
1-096 776 X 10 s cm *. 


The value of fl_ is referred to as a rydberg {1 Ry ~ 13- 60 eV) 
Note that 1 Ry is half the -atomic unit of energy (the hartree 
E : IE =2Ry). 

a a 



said, we refer to the true Rydberg constant as the quantity 
R in Box 21 . The Rydberg constant for the hydrogen atom, 
taking into account the finite mass of the proton, is 
ft = r /(i + m Im ); this is also recorded in the Box. The 

H w a P 

Rydberg constants for other nuclei can be obtained by 
replacing the mass of the proton in this expression by the 
mass of the nucleus of interest. These different expressions 
arise because the electron-proton system rotates around its 
centre of mass, which is slightly shifted away from the position 
of the nucleus by virtue of its finite mass and the electron's 
non-zero mass. 

Further information. See -Bohr atom for the first calculation 
of the Rydberg constant in terms of fundamental constants, 
and "hydrogen atom for the basis of its quantum-mechanical 
deduction. Both entries give further information. 

Rydberg level. An electronic transition in a molecule might 
lift the electron out of the valence-shell orbitals into an outer 
orbital: the state so formed is a Rydberg state, and the electron 
occupies a Rydberg level. An example of this would be the 
excitation of a 2p-electron of the fluorine atom in the fluorine 
molecule into a 3s-orbital, or something higher. 

The Rydberg levels are of interest in so far as the important 
electron is in a very diffuse orbital; so diffuse, in fact, that in 
a diatomic molecule the two nuclei appear to the electron as 
a single nucleus. This implies that the Rydberg electron is 
only very weakly coupled to the nuclear framework, which is 
therefore able to rotate without dragging the electron round 
with it: see "Hund's coupling case (d). Rydberg electrons are 
characterized by small -quantum defects: since they are so 
diffuse they hardly interact with the inner electrons and their 
wavefunctions resemble those of the "hydrogen atom. 

Further information . Rydberg levels and states are discussed in 
some detail in §10.3 of King (1964) and gVI.Bof Herzberg 
(1950). For a thorough discussion see Duncan (1971). 



s 



Schrodinger equation. The Schrddinger equation, which by 
one of those rare coincidences is named after him who did 
indeed discover itt, is a differential equation whose solution 
is the "wavefunction for the system under consideration. This 
implies that it is of central importance, for once we have the 
wavefunction all the properties of the system are, in principle, 
predictable, because the structure of quantum mechanics tells 
us how to elicit the Information. The application of quantum 
mechanics to physical systems therefore boils down to solving 
the appropriate Schrddinger equation, and realizing that the 
mathematical function which is the solution is the wave- 
function for the system. 

Unfortunately the Schrodinger equation is not a simple 
algebraic equation [like x 2 = 2), but, as mentioned above, it is 
a differential equation. Except in a fairly small number of cases 
such equations are very difficult to solve. That, however, is not 
of much significance: what matters is that we believe we have 
the equation which, in principle, is the correct equation for tin- 
description of Nature. (Actually that is not really true, for the 
Schrodinger equation ignores relativity. Therefore it is only an 
approximation, bearing a similar relation to a correct des- 

t 'While visiting Paris he (Victor Henri) received from Langevin a copv 
of "the very remarkable thesis of de Broglie"; back in Zurich and having 
not very well understood what it was all about, he gave it to Schrddinger, 
who after two weeks returned it to him with the words: "That's 
rubbish". When visiting Langevin again, Henri reported what 
Schrodinger had said. Whereupon Langevin replied: "I think 
Schrodinger is wrong; he must look at it again". Henri, having returned 
to Zurich, told Schrodinger: "You ought to read de Broglie 's thesis 
again, Langevin thinks this is a very good work"; Schrodinger did so and 
"began his work".' Max Jammer (1966, p. 258). 



cription as Newtonian mechanics bears to Einsteinian. This is 
a difficulty which has been partly removed— see the -Dirac 
equation— but minor fundamental difficulties have in the past 
bred cataclysm.) 

The Schrodinger equation is a second-order linear differ- 
ential equation in space coordinates (it contains terms such as 
d 2 /dx 2 ] and a first-order differential equation with respect to 
time. Various forms of it are illustrated in Box 22. Written in 
its full form it should be clear that It is not a wave equation, 
for such an equation has second-order derivatives with respect 
to time. It may be regarded instead as a type of diffusion 
equation; it is not unreasonable that the evolution of the 
'wavefunction' in time should be akin to a diffusional process. 
This point is of considerable significance, for the diffusional 
form of the Schrddinger equation means that it is possible to 
to interpret the wavefunction in terms of a probability of 
discovering a particle in various regions of space (see Born 
interpretation in the section on "wavefunction). Had the 
equation been a true wave equation this interpretation would 
have been untenable. The time-dependence can often be 
shaved off by the method of separation of variables (see 
Question 2 and Box 22), and then we are left with the time- 
independent Schrodinger equation (see Box 22), which is of 
the same form as an equation for a standing wave. It is from 
this form of the equation that the name 'wave mechanics' 
derives. 

The time-independent Schrodinger equation may be inter- 
preted as an equation for the curvature of the wavefunction, 
and bearing this in mind enables one to anticipate some of the 
features of its solution. The second derivatives 9 2 0/3x 2 , etc. 



207 



208 



Schrodingcr equation 



BOX 22: The Schrodinger equation 

Time-dependent form: Hty= ih0 */9 1 ) 

Wis the -hamiltonian. 

Time-independent form: if Wis independent of time, ^ may 
be written 

#= if/ exp(-i£ tfh), 

where \j/ is independent of f and satisfies 

Typical form of equation: 

One-dimensional system; massm in a potential V{x\: 



or 



(f ) < f 



+ l/(x}^U)=F0{x) 



dx 2 



V(jf)l^W = 0. 



Three-dimensional system; massm in a potential U(r): 
V 2 #) + lW<r) = £^(r). 



where V 2 is the "laplacian. 

Separation of variables. Write ^ = >jjd, where i^ is a 
function of position and a function of time. Let H be 
independent of time. Then H^ - ih\If becomes 

0W0 = ih\#, or (1/^tf if» = ith/6)&. 
The left-hand side is a function of position, not time, and if 
x is varied the right-hand side is invariant. Therefore l.h.s. = 
constant = r.h.s. 

Hence 6 = exp{-i£f/h) and H\jj = E\j/. 



are what we normally interpret as the curvature of the 
function in elementary calculus, and we shall employ this 
interpretation here. Note that when ip" fas we sna H denote the 

second derivative) is positive the curvature of \p is like \. S , 

and when it is negative the curvature is like /"~N . The 
essential feature to note is that the magnitude of the curvature 
(the sharpness with which the curve bends) increases as the 
total energy £ exceeds the potential energy V[x). This differ- 





E<V 


E>V 


y>>o 


w 


r\ 


1p<Q 


r\ 


w 



FIG. S1. Curvature of the wavefunclion at a point for different signs 
of t}> and B— V at thai point. 



ence £ — VM is just the classical kinetic energy at the 
point x, and so we see that the "kinetic energy and the 
curvature are proportional. Note too that the sign of the 
curvature depends on the sign of the function \p itself: if E is 
everywhere larger than V, the curvature has the sign of i> at 
each point (Fig. S1). It is amusing to follow through the 
implication of this for a free particle where V is constant and 
E> V. Suppose we consider a point where \p> 0, then the 
curvature is negative and so the function droops down towards 
zero like /* — \ (Fig, S2). Sooner or later this droop causes it 
to fall through zero and become negative. The hitherto droop- 
ing function acquires a positive curvature (because still E> V 
but i/»< 0) and so begins to curl up towards the value zero. 
This value it crosses and then again begins to droop back down. 
The dependence of the curvature on the function therefore 
forces the function to swing backwards and forwards across 
the axis, and so to describe a harmonic wave. It is also amusing 
to note that the rapidity with which it swings from positive to 
negative values increases as E exceeds V: therefore the wave- 
length of the motion decreases as the kinetic energy increases 
(see 'kinetic energy and the °de Broglie relation). If the 
potential depends on position the wavelength is not a constant 
(and not really defined}, but these arguments may be extended 
to account qualitatively for the form of wavef unctions for 
electrons in atoms and molecules. 



Schrodinger equation 



209 



f 



>0 




r 




FIG. S2. Propagation of a wave when E > V. 

The only adequate way of accounting for the quantitative 
form of the wavefunction in atoms and molecules is to solve 
the equation mathematically. The immediate problem that 
one encounters is that there exists an infinity of solutions for 
a second-order differential equation. The essential point at 
this stage is the recognition that only some of the solutions 
satisfy the stringent requirements of the Born interpretation: 
when boundary conditions are imposed (that is, when one 
states what conditions the function must satisfy at some 
point of space) only a few solutions are acceptable. The 
immediate consequence of this is that bounded systems are 
quantized. 

Quite often it is impossible to find analytical solutions to 
the Schrodinger equation, or at least to find analytical 
solutions that are not too complex to use. Under these 
circumstances (which include the immensely important cases 
of the structure of atoms and molecules) it is necessary to 
resort to approximate methods: these include "perturbation 
theory, "variation theory, and the method of "self -consistent 



fields. Time-dependent perturbation theory is one way of 
dealing with the Schrodinger equation. Some of the standard 
solutions of the Schrodinger equation are discussed under the 
appropriate headings (see -particle in a square well, "harmonic 
oscillator, "angular momentum, and -hydrogen atom). 

The fact that the Schrodinger equation is a linear differ- 
ential equation implies the validity of the "superposition 
principle and all that flows from it. 

Questions. T. What is the importance of the Schrodinger 
equation? Why is the time-dependent form not a true wave 
equation, and how may it be manipulated into a form that 
looks like a wave equation? What basic local property of the 
wavefunction does the equation determine? Demonstrate 
qualitatively and then quantitatively that the wavefunction 
for a free particle is a wave of constant length. What is the 
connexion between the length of this wave and the linear 
"momentum and "kinetic energy of a particle described by 
such a wave? Sketch the form of the wavefunction for a 
particle of energy E in a potential field that decays linearly 
with distance. Can a particle be described by a wave in which 
at some point the energy E is less than the potential energy V 
at that point? Sketch the form of a wavefunction for a particle 
in which the potential rises linearly with distance (and crosses 
the point V{x) = E). Sketch a number of possible wave- 
functions for a 'particle in a square-well potential, and in a 
"harmonic potential {try to do this without looking at the 
answers, which will be found in the appropriate sections; if 
you cannot do it, attempt to interpret the answers there in 
terms of the discussion in this section), 
2, Show, by the method of separation of variables, that the 
time-dependent equation can be separated into an equation 
for the time-dependence and an equation for the spatial 
dependence, and solve the former. Proceed by attempting to 
express the function Mx, f) as the product 0{t) \p{x), and 
inserting this in the time-dependent equation. Divide through 
by &4j, and realize that one block of terms depends only on x 
and the other depends only on t. Deduce that these two terms 
must each be equal to some constant, which write E (on the 
grounds that the constant has the dimensions of energy). Then 
solve the equation for the time, and compare your answer with 
that in Box 22. 



210 



second quantization 



Further information. See MQM Chapter 2 for a way of setting 
up the Schrodinger equation, and Chapter 3 for the method of 
obtaining some of its solutions in a number of important cases. 
A good account is also given in Pauling and Wilson (1935) and 
in Kauzmann (1957). All books dealing with quantum theory 
must, except for the most abstruse, mention the Schrodinger 
equation. For Schrddinger's original papers see Schrodinger 
(1926). See Chapter 5 of Jammer (1966) for a fascinating 
commentary, and in an English translation in Shearer and 
Deans (1928). The Schrodinger equation is only one of a 
number of possible equivalent formulations of quantum 
mechanics, and is not always the simplest to use, especially in 
formal manipulations. Therefore see also -matrix mechanics. 

second quantization. Those who find quantization a 
sufficiently difficult topic will be distressed to encounter 
second quantization, and may have visions of a continuation 
yet more subtle than Chinese boxes. But second quantization is 
as far as things have gone, and is a device that enables one to 
conjure elegantly with problems involving many particles, 
including the problem of the electromagnetic field. It is a 
mathematical artifice; but is that not possibly true of all 
mathematical descriptions of Nature? 

The mode of thought that leads to the introduction of the 
idea is as follows. First quantization, which in our naivety we 
have referred to elsewhere simply as quantization, replaces 
observables by "operators, and the behaviour of a system, and 
the results of experiments, are calculated by allowing these 
operators to operate on the wavefunction obtained as a 
solution of the "Schrodinger equation for the system. Thus 
the dynamical functions have been replaced by operators 
operating on a function \}/(x). But suppose by analogy with 
this development we interpret the function \J/{x) as an 
operator on something; then we have gone beyond our formal 
procedure for quantization and are in the artificial realm of 
second quantization. In this realm it is discovered that the 
operator ^/*M operates (on something) to create a particle 
atx, and that the operator \p{x') operates to annihilate a 
particle at x'. It should be possible to appreciate that this 
power of summoning and dismissing particles provides a means 
of setting up equations that enable the quantum-mechanical 
properties of many-body problems to be calculated. Thus 



second quantization does not introduce a revolution into 
physics (as did first quantization), but it does introduce a new 
technique of calculation, and a new language— a language that 
includes words such as -photons, "phonons, "polarons, 
"excitons, magnons, and rotons. 

Further information. An introduction to the ideas of second 
quantization will be found in a set of lecture notes by Atkins 
(1973), which are based on a book by Mattuck (1967). Other 
introductions of increasing sophistication will be found in 
Ziman (1969), Roman (1965), Davydov (1965), Bogoliubov 
and Shirkov (1959), and Schweber (1961). See also Kittel 
(1963). 

secular determinant. In the construction of a "molecular 
orbital by the method of "linear combination of atomic 
orbltals one attempts to express the orbital as a sum 
c ty + c \p + . . ., where the i^s are the atomic orbitals on 
atoms a, b, etc. and the cs are the coefficients to be modified 
until the best set of values is found: this is done by seeking the 
set that gives the lowest energy. Instead of idly toying with 
random different values of the cs until mental decay super- 
venes, it is desirable to have a short, sharp method for finding 
the best set, and this is the role played by the secular 
determinant. 

In the Questions you are asked to show that the best com- 
bination of orbitals is given by one of the solutions of a set 
of simultaneous equations in which the coefficients c are the 
unknowns (see Box 23). These are the secular equations. In 
common with other sets of simultaneous equations they 
have a non-trivial solution when the determinant of the 
factors of thecs disappears (the trivial solution corresponds 
to all the cs themselves vanishing). This determinant is the 
secular determinant. See the Box. The secular determinant 
vanishes for N values of the energy of the system, if there are 
N atomic orbitals contributing, and the lowest roof may be 
identified with the lowest energy of the system, that can be 
attained on this model. The values of the cs corresponding to 
this root may be found by the normal methods (brute force, 
or intelligence via the method of cof actors, see Box 23) and 
give the best wavef unction for the system. The other (A/— 1) 
roots may be identified with higher-energy orbitals. for the 



secular determinant 



211 



BOX 23: The secular determinant 

Direct approach. Write the Schrodinger equation H\p=&ip 
in terms of the expansion 

fit 

y= 1 

and obtain N N 

£y*/-£ «*.*.- ft 

y - i / - 1 

Multiply from the left by \p. and integrate, to obtain 
N 

£<y[H-&S]= (/=1,2,.../V) (2) 

/-i 
with H j} - fdTfyhtty and S /y = fdT\j/.\L. This is a set of /V 
simultaneous equations for the coefficients c, and possesses 
non-trivial solutions when 



A= det l/A. 
8 



-es y i = o. 



(3) 



Expansion of this fl/X/V determinant, and solution for £ 
leads to the N eigenvalues of H, In general, if N is finite the 
eigenvalues are only approximate. 

To each root & of {3} there corresponds a wavef unction ^ 
expressed as in (1 ). To find the coefficients c. use Cramer's 
rule. (See Margenau and Murphy {1956, p. 313) or Irving 
andMulfineux(1959,p. 269).) Let A, = c./c,; then (2) 
becomes 

iLkfVHfi ~ &S v l = m ft - H n ) (/ = 2, 3, . , . N\. 

Let D be the (Af— 1) X (A/-1 ) determinant formed from 
this H l} - &S.j, and D w {h) be the determinant formed by 
replacing the nth column in D by (£S 2I - tf 2l , &S 3 , - W 31 
. , , &S 



— MffJ- Then according to Cramer's rule: 

., c N /ci. To complete the 



This gives the ratios c 2 /ci , 
determination use 

c i + c\ + 



..+4-1. 



(4) 



This procedure gives the coefficients e. corresponding to 
the root &, and should be repeated for each root of (3). 



Variational approach. Minimize S with respect to variations 
in the coefficients c, 

The condition (3S/c)c.) = is satisfied when 



N 



sty 



(5) 



A = 



(Confirm this by substituting (1), taking the derivative of £, 
and using simple algebra.) This is identical to (2) and the 
procedure is as before. The lowest of the N roots of (2) is 
the minimum value of £. 

Example. Suppose N — 2, then the secular equations are 
Ci Wn - &S n ) + c 2 {H n - &S n ) = ) 
ci (W 2I - &S 2 i) + c 2 (W 22 - £$ 22 » = \ 

and the secular determinant is 

"31 ~~ &S 2 1 H22 — &S22 
The roots of A are the roots of 
<S U S 22 -S 12 S 21 )g 2 + tH,A, + H 2l S n - H 22 S n - «„S 22 )E + 

and the lower is the minimum of S. The coefficients are 
determined from D = W 22 — &S 22 and eqn (4): 

«2 W 22 — &S22) — ~ {H2I ~ SSji) 

and c ^+c| = 1. 

In the special case when S n => S 2 2 = 1; Si 2 = S 2i = the 
solution may be put into the form 

« _ j Wn — H n cot0 ip = i^,sin^ - \p 2 cos8 

\ ri-n + H l2 coxd \p = 1^ cos0 + ^ 2 sin ' 

where = ^arctan[2W, 2 /(W 11 -W M )]. 



molecule, and the corresponding coefficients give the other 
(/V — 1 ) molecular orbitals. 

The secular determinant crops up wherever one has several 
orbitals which one anticipates may interact. If there is a 
'perturbation that can mix one state with another the true 



212 



selection rule 



ground state of the system is best found by a linear sum of 
the two unperturbed states. The energy of the new system is 
given by the lowest root of the secular determinant, and the 
set of coefficients corresponding to that root gives the best 
modification of the wavefunction. One example of this type 
of situation is provided by •configuration interaction, and 
another by the -Huekel method of conjugated molecules. 

The name secular originates in the appearance of the same 
kind of determinant in classical mechanics, and especially in 
celestial mechanics. A secular variation in the motion of a 
body, and in particular the orbit of a planet, is one that 
gradually develops over a long period of time {saeculum: 
Latin for age or generation) as opposed to one that varies 
rapidly or periodically and is not cumulative. Why on earth, 
if you will forgive the allusion, does this have anything to do 
with molecular structure? The answer lies in the fact that 
variation theory may be related to perturbation theory, and 
that the perturbations of interest to variation theory are 
those that accumulate to give rise to a set of orbitals with 
well-defined and constant separation. This shows up 
especially clearly in degenerate-state perturbation theory, but 
that is a subject that deters unless it is called by some other 
name. One name that ought not to deter is "molecular-orbital 
theory, for when homonuclear systems are considered the 
molecular-orbital method is equivalent to degenerate-state 
perturbation theory. This can be appreciated by realizing 
that in the absence of interaction between the atoms all the 
atomic orbitals {of the same quantum numbers) which later 
are to be combined have the same energy (are -degenerate). 
The effect of the interatomic interaction is like a perturbation 
on the degenerate systems. If we imagine putting the atoms in 
a molecular conformation, forbidding interaction, and then 
gradually turning on the interaction, we can appreciate that 
the perturbation gradually accumuiates and the different 
linear combinations of atomic orbitals diverge until they 
attain the separations characteristic of the molecule. Thus we 
are really considering a strong secular perturbation on the 
atomic orbitals, and the separation of the levels can be found 
by the application of the secular determinant. 

The language of ordinary "perturbation theory also draws 
on the word secular. The expression for the energy to second 
order in some perturbation consists of a part involving only 



the original state of the system (see Box 16 on p. 172) and a 
part involving "virtual transitions to excited states. The former 
terms represent the effect of a secular term, or secular 
perturbation, whereas the latter represent the effect of the 
non-secular terms. 

Questions. The -variation method considers a linear com- 
bination of orbitals of the form i/i = c a ^ a + c b ^ b for a two- 
orbital system and then calculates the minimum value of 
& = JdT^H\p/fdT\}j\p. Show that the extremal values of this 
expression correspond to the solutions of tfie two simultaneous 
equations |W 3o - &SJc a + («„„ - S>SJc b - and 

are various integrals. Set up and solve the secular determinant 
for this problem, and find expressions for the two sets of cs 
corresponding to the lower and higher energies. Apply this 
calculation to H 2 by identifying ^ a and ip b with Is-orbitals 
on nuclei a and b respectively, taking S aa = S bb = 1 , 
H = W hh = a, and H . = tf h = j3. Suppose that S = 0. 

aa bb lib ba . lM 

Now generalize the calculation to an orbital of the form 
c \1/ +c \b +... c„ 0„ and show that the solution of the 

a T ab T b IV Iv 

variational problem leads to an N X N determinant. Approxi- 
mations are usually made as to the values of the integrals 
involved in these secular determinants: this is the realm of the 
-Huckel method and its analogues, and you will find more 
problems set there. 

Further information. See MQM Chapters 7 and 10; the former 
deals with degenerate-state perturbation theory and the latter 
with the application of the secular determinant to various 
aspects of molecular structure. Further details will be found in 
§9.6 of Coulson (1961), §2.1 of Streitweiser (1961), §Vl.24 
of Pauling and Wilson (1935), §7band§11bof Eyring, 
Walter, and Kimball (1944), Pilar (1968), and McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972). See -Huckel 
method, 'configuration interaction, and "perturbation theory. 

selection rule, Spectral lines result when a system makes a 
transition from one state to another of different energy. All 
lines in a spectrum can be related to the difference between 
the energies of states of the system (each line can be expressed 
as a combination of "terms), but not all possible pairs of states 
give rise to spectral transitions: some transitions are allowed 



and some are forbidden. Selection rules tell us which are 
allowed and which are forbidden. They are generally quoted 
in terms of the changes that may occur in a "quantum number, 
but sometimes they are rules about the way that the symmetry 
of the state may change. Occasionally one encounters the term 
gross selection rule: this refers to a property that a molecule 
must possess in order for the remaining selection rules to come 
into operation. Selection rules for a variety of transitions are 
shown in Box 24. 

There are various ways of understanding why the selection 
rules govern as they do. The gross selection rules refer to the 
fact that the molecule must have some way of interacting with 
the electromagnetic field: the presence of a permanent dipole, 
for instance, means that the electric field of a passing light 
beam may accelerate the molecule by exerting a torque and so 
induce a -rotational transition. The other selection rules can 
generally be understood in terms of the possession by the 
photon of a definite intrinsic angular momentum (a -photon 
has -spin,. On absorption the photon is annihilated but 
angular momentum must be conserved, and this momentum 
appears in the electrons or in the nuclear framework. Rules 



selection rule 



213 



BOX 24: Selection rules 

Atoms 

Electric dipole transitions: 
A/ = 0,±T;butJ = 04*J = 
9— >u;g4*g, u4+u (Laporte rule) 
AS=0 

4t = 0,±1;buti = 0-ki = o 
Magnetic dipole transitions: 

A/ = 0, +1; but J = 04*^ = 

g — *g, u — >u;g+>g 

AL = 0, ±2. 
Electric quadrupole transitions: 

A/=0,+1,±2;butJ = 04*./=0 

g— *g, u— >u;g4*u 

AL = 0,+V . ±2; but L = 4* L = 0. 



n /.^-coupling. 



>A =0: 



Molecules 

Electric dipole transitions: 
electronic transitions 
AJ=0, ±V,butJ = Q+>J = 
+ -*-; + 4*+,-4*- 
g— *u;g4*g, u4*u 
s — * s, a — * a; s 4+ a 
case (a): AA = 0, ±1; but for A= 
T— *!?, 2T- 
AS=0 
A2=0 
cases {a} & (c): AJ2 = 0, ±1; but 12= 04* £2 = if AJ = 
cases (b) & (d): AN = 0, ±1; but in (b) A= 04* A= 
for AN = 0. 
vibrational transitions: 

absorption (i.r.): dipole moment must change along 
"normal coordinate 
Af=±1. 

Raman: polarizability must change with vibration 
Ap=±\. 

vibration-rotation transitions: 

+2 S-branch (Raman, Stokes) 

+ 1 R-branch (i.r.) 

Q-branch t (i.r.) 

'—1 P-branch (i.r.) 

[ ~2 O-branch (Raman, anti-Stokes). 

t Only if molecule has component of 
a.m. about axis. 

rotational transitions: 

absorption (microwave): molecule must possess 

permanent dipole moment 

Aj=+1, AK =0. 
Raman: anisotropic polarizability of molecule 
Aj = ±2 (Stokes: +2; anti-Stokes: —2). 



Ac = ±1; AJ = 



214 



self -consistent field 



such as M = ±1 or AJ = ±1 reflect this conservation of 
angular momentum, and are discussed further under 'electric 
dipole transitions and in the sections on the different types 
of spectra (see also "magnetic dipole and electric quadrupole 
transition for a variant of this rule). 

AH the rules may be deduced from an examination of the 
form of the transition dipole moment (see -electric dipole 
transition). One very important way of examining this 
moment is by "group theory: those who know about group 
theory and irreducible representations should remember that 
the product of the irreducible representations of the initial 
state, the final state, and the transition moment operator 
must contain the totally symmetric irreducible representation 
for the transition not to be forbidden (Box 4 on p. 33). Those 
who do not know enough group theory to understand this 
important rule should learn some as soon as possible. 

Further information. See MQM Chapter S for a discussion of 
the group-theoretical basis of selection rules, and Chapters 
8 and 1 for their application to all the types of atomic and 
molecular transitions. Other group -theoretical discussions 
will be found in the references in the entry on group theory. 
The applications to spectra are also described under 
"electronic, -vibrational, -rotational, and "Raman spectra, and 



detailed accounts will be found in Barrow (1962), Whiff en 
(1972), King (1964), and Herzberg (1944, 1945, 1950, 1966). 

self -consistent field. The self-consistent field (SCf ) method 
of calculating atomic and molecular structures was originated 
by Hartree, improved by Fock, and used by almost everyone. 
The basis of the method is to guess the °wavef unctions for all 
the electrons in the system. Then one electron is selected and 
the potential in which it moves is calculated by freezing the 
distribution of all the other electrons and treating them as the 
source of the potential. The Schrodinger equation for the 
electron is solved for this potential, and so a new wavefunction 
for the electron is obtained. This procedure is repeated for all 
the other electrons in the system, using the electrons in the 
frozen "orbitals as the source of the potential. When the cycle 
is completed (Fig. S3) one possesses a set of orbitals for all 
the electrons of the system, and in general these will differ 
from the original, guessed set. Now the cycle is repeated, but 
improved wavefunctions generated by the first cycle are used 
as the initial guess: a complete cycle generates a new set of 
improved functions. This sequence is continued until passing 
a set of orbitals through the cycle leads to no change: the 
orbitals are then self -consistent. 



Storf here 



FIG. S3. The self -consistency 
cycle. 




Bid of game 



self -consistent field 



215 



1. The Hartree method takes the view that the atom or 
molecule can be represented as a product of orbitals, one 
for each electron. Therefore it guesses a set of functions, and 
sets the self-consistent machinery in operation; after a number 
of cycles the solution has become stable, and so one has a 
function (still a simple product of one-electron orbitals) that 
describes the structure of the system, and a set of orbital 
energies from the solution of the Schrodinger equation. The 1 
calculation is actually done with the Hartree equations (see 
Box 25) in which the potential experienced by one electron is 



BOX 25: Self-consistent fields 

Hartree equations 



where the Coulomb operator is 



/- i 



1 



J f m ~ /dr 2 4** (2) $. (2) (eV47reor 12 ). 

Sum/ runs over occupied orbitals. 
Hartree-Fock equations (closed shell) 

[W corB (1) + 2j>.(1) -£ #f (1)] tyi, = e^.m, 

where the exchange operator is defined through 
*,■ ( 1 ) lf», < t ) = Jdr 2 tyf (2) }jj. m tftAiKtfn) tym 

With J,y = /dT, ^'(1)^(1 ) 1^.(1} 

as the Coulomb and exchange integrals. 



the sum of its interactions with all the other electrons. The 
presence of the electron is not allowed to distort the electron 
clouds locally: its effect is to distort the orbital as a whole. No 
possibility of the other electrons tending to keep away from 
the position of the electron of interest is admitted: electron 
correlation effects are ignored; see Fig. C3 on p. 43. 

2. I n the Hartree-Fock method the self-consistent field 
procedure takes into account the indistinguishability of the 



electrons as required by the -Pauli principle. That is, it allows 
for the possibility of "exchange. To do so it takes as the 
wavefunction a 'Slater determinant, and then enters the 
self-consistency cycle of Fig. S3. The potential experienced by 
each electron is the Coulombic potential modified by the 
"exchange energy {which is the correction of the Coulombic 
repulsion energy required in order to take into account the 
tendency of electrons with like spins to avoid each other). The 
Schrodinger equation is cast into the form of the Hartree-Fock 
equations (Box 25), where the first part of the potential is the 
uncorrected Coulombic interaction of the electron with all the 
other electrons in their frozen orbitals, and the second part is 
the exchange-energy correction. The Hartree-Fock (HF) 
method also neglects the -correlation energy. The unrestricted 
Hartree-Fock (UHF) method allows more freedom to the form 
of the orbitals by permitting the spatial form of the orbital to 
depend on whether the electron has an a or a (3 "spin. 

It is normal for the atomic and molecular orbitals used as 
a starting point in SCF calculations to be -linear combinations 
of atomic orbitals, and the accuracy of the calculation is 
severely curtailed if the functions chosen are too inflexible; 
this might occur if too few (too small a basis set] have been 
chosen. A convenient set of orbitals with which to commence 
a calculation are the "Slater atomic orbitals. The evaluation of 
molecular integrals is considerably simplified if -gaussian 
orbitals are used, but more of these must be used if the atomic 
wavef unctions are to be at all reasonably represented. When 
the labour of an SCF calculation appears to be too great, or is 
actually found to be too great, approximations are introduced 
in a more or less rational fashion: such methods constitute the 
semi-empirical SCF calculations (see -Huckel method) as 
opposed to the a priori or 'ab initio SCF calculations which 
begin from scratch and proceed without approximation (apart 
from the approximations inherent to the HF scheme). In all 
cases the calculations can be improved by permitting -con- 
figuration interaction. 

Questions. Outline the sequence of calculations involved in a 
self-consistent field calculation. When does the cycling pro- 
cedure cease? What is the basis of the Hartree method? What 
does it neglect? What is the basis of the Hartree-Fock method, 
and why is it an improvement on the Hartree method? Why is 



216 



sequence 



the HF-SCF method unable to provide exact atomic wave 
functions and energies? What is the UHF-SCF method? What 
interpretation could be put on the letters HF— SCF— LCAO— 
MO, and what process of computation would you understand 
by it? What is the difference between an ab initio SCF cal- 
culation and a semi-empirical calculation? What errors are 
introduced by using too small a basis set, and what is meant 
by a 'basis set' in this context? Show that it is necessary to 
deal with electron repulsion and exchange integrals involving 
orbitals located on up to only four atomic centres. Why is it 
unnecessary to invent methods to deal with 5-centre integrals 
in HF calculations? 

Further information. See MQM Chapter 8. A simple intro- 
duction to the methods of atomic and molecular energy 
calculations is given by Richards and Horsley (1970) and by 
McGlynn, Vanquickenborne, Kinoshita, and Carroll {1972}. A 
classic account of atomic-structure calculations, which 
illustrates the headaches which Hartree must have suffered 
before electronic computers were available, is described in 
Hartree (1957), Both Richards and Horsley and McGlynn 
etal. work through sample calculations. The semi-empirical 
methods are described under "Huckel method. 



sequence. First review and be quite clear about the meaning 
of a "progression in the "vibrational structure of the "electronic 
spectra of molecules. A sequence is a series of lines that have 
in common the same value for the difference of the vibrational 
quantum numbers for the upper and lower electronic states. If 
the upper vibrational quantum number is V and the lower V , 
then the lines that have v' — v" = form one sequence, those 
with V — y" = —1 form another, those with j/ — 1>" = + 1 a 
third, and so on (see Fig. S4). 

All the lines of one sequence would lie at the same 
frequency if the vibrational energy levels in both the 
electronic states were evenly spaced, but the -anharmonicity 
of the vibrations destroys the even separation of a truly 
"harmonic oscillator, and so the separation of the lines of a 
sequence yields information about the deviation of the 
molecular potential -energy curve from an ideal parabolic 
form. 



5 */ 



6u=\ 



4 

3 

2 



I 



6v-0 



V 



FIG. S4. Two sequences. 

Questions, What is a sequence, and how does it differ from a 
progression? What information is it possible to extract from 
the positions of the lines in a sequence? When may all the lines 
of a given sequence be coincident? Would you expect the 
intensity of the lines in a sequence to be the same; if not, what 
is a major influence on the intensity? Given that the vibrational 
energy of the ground state depends on v" through the 
expression gj'V + fl + x"jJ^ + \) % . with a similar 
expression for the excited electronic state vibrational energies, 
calculate the frequencies of the 0, — 1, and +1 sequences. 

Furtfier information. See MQM Chapter 10. Detailed infor- 
mation about the appearance, analysis, and formation of 
sequences will be found in Barrow (1962), Whiff en (1972), 
Gaydon [1968), and Herzberg (1950, 1966). 

series. The frequency of any spectral transition may be 
expressed as the difference of two "terms, each term 
representing the energy of a state of the atom or molecule. 
Let us denote the terms T , where n is some index which 



is generally identified as a "quantum number or a collection 
of quantum numbers for the system. The frequency of each 
transition from a state n to a series of other states n' is simply 

~^n> ~ T n' and tne series of " nes ' m tne spectrum for a fixed n 
and changing n is called a spectral series. 

Some of the most famous spectral series occur in the alkali 
metals and involve transitions of the single valence electron. The 
transitions in which an electron in a p-orbital drops down into 
the ground-state s-orbital (see Fig, S5) gives a series of lines 
known as the principal series (and hence the notation 'p' for 
the orbitals involved}; the series formed by the light emitted 
as the electron falls from some upper s-orbital into the lowest 
p-orbital constitutes the sharp series {and hence V); the 
decay of electrons from the upper excited d-orbitals falling 
into the lowest p-orbital gives a diffuse series {thus it looks, 



u 
S 

I 

o 




5000 



15000 t 



25000.9 



s 

2 



iSOOO 



FIG. S5. S, P, D, F series in sodium. 



singlet and triplet states 217 

and hence 'd'); and as electrons in f-orbitals drop to the 
lowest d-orbital so is generated the fundamental series (and 
hence T). The transitions are illustrated in Fig. S5. 

Further information. See MQM Chapter 8 for more infor- 
mation about series. The structure of atomic spectral 
series is described by White (1934), King (1964), Herzberg 
(1944), Whiffen (1972), Kuhn (1964), and Condon and 
Shortley (1963). The "selection rules that led to the con- 
struction of Fig. S5are described in that section. Molecular 
series are discussed in King (1964), Barrow (1962), Whiffen 
(1972), and Herzberg (1950, 1966). 

singlet and triplet states. In a singlet state the net 'spin 
of a many-electron system is zero {S= 0). In a triplet state 
the net spin is unity {S = 1). The spin "angular momentum 
•vector may have a series of projections on a selected axis. 
These projections are distinguished by the quantum number 
M s which can range in unit steps from S down to S. It 
follows that /W„ may take on three values (M„ = 1, 0,-1) 

o o 

when S = 1, but only one value {M s = 0) when S = 0: hence 
the names triplet and singlet. 

The distinction is easiest to see in the case of a system 
composed of two electrons. As each electron can have a pro- 
jection m s = +j or — j (which we denote a or 0) the com- 
bined system can be in any of the four states a(1 )u{2), 
<*{1 )0{2), £(1 )a(2), and j3( 1 )j6(2). The middle two choices do 
not correspond to a resultant spin vector of fixed length 
because the a and vectors can make any azimuthat angle to 
each other. If we specify the azimuth of one with respect to 
the other we shall get a definite resultant: if a{1) and (3(2) 
are in phase they give a resultant corresponding to S = 1; if 
they are 180° out of phase their resultant is zero and 
corresponds to a state with S = (see Fig. S6). It is worth 
emphasizing that when we say that spins are 'paired' in a 
singlet state we mean not only that one has a spin and the 
other but also that they are relatively oriented so that they 
point in opposite directions. (In the M s = state of the triplet 
one electron has a spin and the other )3 but their resultant is 
not zero.) One can show from quantum mechanics that the 
appropriate form of the spin function for the triplet (in-phase) 
state with M s = is a{1 )(3(2) + 0( 1 )a(2), and for the singlet 



218 Slater atomic orbitats 





FIG. S6. Triplets and singlets: 
note the different phase of a and 
in the M =0 state of the singlet 
and the triplet. 



(out-of -phase) state it is <*(1 )0(2) — 0(1)0(2), The other two 

states of the triplet (M s = ±1) area(1)a(2) and j3< 1)j3(2). 

It is possible to convert a singlet term into a triplet (and 
vice versa} by making one electron spin precess faster than the 
other. This may be brought about by applying different mag- 
netic fields to the two spins so that their Larmor frequencies 
(see "precession) differ: an in-phase (triplet) orientation 
is thereby gradually turned into an out-of-phase orientation 
(Fig. S7). A field from a laboratory magnet cannot effect the 
interconversion (which is known as intersystem crossing, ISC) 
because it affects both spins equally {the magnet's field is 
homogeneous on a molecular scale). A magnetic field arising 
from within the molecule may be able to rephase the spins. 
For example, the "spin-orbit coupling to one electron might 




FIG. S7, Relative re-phasing of the spins lead to triplet-singlet inter- 
conversion. The hatching denotes a region of a different spin-orbit 
coupling. 



differ from that to the other, and as this interaction is 
magnetic, we have a situation in which the Larmor frequencies 
differ. This is the reason why molecules containing heavy 
atoms (with large spin-orbit couplings) are efficient at inter- 
system crossing (see "phosphorescence). 

The interconversion of ortho- and para-hydrogen (see 
°nuclear statistics) can be brought about by paramagnetic ions 
because the singlet and triplet phasing of the nuclear spins can 
be interconverted by the inhomogeneous field which such an 
ion may generate if it is closer to one nucleus than the other. 

Further information. See Chapter 6 and Chaper 10 of MQM 
for a more detailed discussion of singlets and triplets, and their 
interconversion. The role of the triplet state in chemistry, and 
its detailed quantum mechanics, is discussed in McGlynn, 
Azumi, and Kinoshita (1969), who give many references. The 
photochemical consequences of the differences between singlet 
and triplet states are discussed by Wayne (1970) and Calvert and 
Pitts (1966). The difference is taken up in -Hund rules and 
"spin correlation, which you now should see. A manifestation 
of singlet-triplet ISC is "phosphorescence. 

Slater atomic orbitals. Atomic orbitals in many-electron 
atoms have a complicated dependence on position which can 
be represented accurately only be listing their amplitude 
numerically. For many purposes it is desirable to have an 
analytical function rather than a table of numbers, and the 



Slater determinant 



219 



Slater atomic orbitals are analytical functions based on the 
numerical results, but designed to reproduce them with 
moderate accuracy. A set of simple rules has been devised 
which enable the Slater-type orbital (STO) to be written for 
any electron in any atom. 

Each orbital has a radial dependence given by Mr"' " 1 
exp(— £r) and an angular dependence given by the "spherical 
harmonics corresponding to the appropriate values of the 
quantum numbers E andm £ (see Table 23). The rules for 
finding the effective principal quantum number n* and f are 
as follows. 

t. n* is related to the actual principal number n by the 
correspondence n •— > n* using the rule 1 — »- 1,2 — *■ 2, 

3 -* 3, 4 — > 3-7, 5 -+ 4, and 6 -* 4-2. 

2. {" is related to the effective atomic number Z _ by 
i - % ff '' 1 > z eff is related to the true atomic number by 
^gtf — Z — a; a is the screening constant. 

3. The screening constant is calculated by classifying 
atomic orbitals into the following groups: (Is); (2s, 2p); 
(3s, 3p); (3d}; (4s, 4p); (4d); <4f); (5s, 5p); <5d); . .'..tristhe 
sum of a number of contributions arising from each group, and 
is calculated as follows. Let the atomic orbital of interest be in 
a group X and let it have a principal quantum number n. The 
contribution from the other electrons present is 

(a) from electrons outside group X (that is, to the right of 
X in the list): 0; 

(b) from electrons in group X; 0-30 if the electron is Is but 
0-35 from any other electron in X; 

(c) if the electron of interest is/is or np 

(i) for each electron with principal quantum number 
number n— 1: 0-85; 

(ii) for each electron with principal quantum number 
n-2, n-3, . ..: 100; 

(d) if the electron of interest is nd or of, for each electron 
in a group preceding X in the list: 1-00. 

Slater-type orbitals for the valence orbitals of the first-row 
atoms are given in Table 21. 

Inspection of the form of the Slater orbitals reveals a 
serious defect: they possess no radial nodes. One consequence 
of this is that they are not "orthogonal. They may be made 



mutually orthogonal by the Schmidt orthogonal ization pro- 
cedure, and so this defect can be overcome (see "orthogonal 
functions for the procedure). 

Questions. Deduce the form of the Slater-type atomic 
orbitals for the Is-orbital in H, 2s in Li, 2s and 2p in C, N, O, 
and F, and the 3d in Fe 2 * and Fe. Find the -normalization 
constant for the general STO. Find the mean radius of the 
electron distribution in each of the orbitals just set up. Show 
that it is possible to choose a sum of the 1s and 2s STOs that 
is orthogonal to the Is-orbital: this is the Schmidt procedure 
for orthogonal izing 2s to 1s, and it may be extended to 
orthogonal ize 3s to both Is and 2s: do so (see "orthogonal 
functions). What effect does orthogonal ization have (in this 
case) on the number of radial nodes, and for the C atom, the 
mean radius of the 2s-orbital? 

Further information. See MQM Chapter 8 for a brief dis- 
cussion. A useful discussion of Slater-type orbitals is given 
in §11.8 of Coulson (1961), Murrell, Kettle, and Tedder (1965), 
and McGlynn, Vanquickenborne, Kinoshita, and Carroll 
(1972). The last, in Appendix B, give many references to the 
expression of "self-consistent field orbitals in terms of sums of 
STO's and a table of orbitals. For the Schmidt orthogonal- 
ization see "orthogonal functions. "Overlap integrals involving 
Slater atomic orbitals are referred to under that heading. 

Slater determinant. According to the "Pauli principle the 
wavef unction for a system of electrons must change sign 
whenever the coordinates of any two electrons are inter- 
changed. It follows that a simple product of the form 
0f <1}y^ (2) i^(3) ... i^(W), where electron 1 occupies 
orbital \j/ g with spin a, and so on, is inadequate. It is possible 
to ensure that a product of this form does satisfy the Pauli 
principle by writing it as a determinant: 



(1W«) V4 



<<D ^ a (2) ^(3) 

Km *J<2) tffoi 

^(1) ^(2) ^ a (3) 



1#U) ^(2) ^(3) 



i/£(/V) 



$*m 



220 



spectroscopic perturbations 



Expansion according to the rules of manipulating determinants 
leads to N\ terms, half occuring with a +ve sign and half with 
-ve. The factor {1//V!} M ensures that the determinantal wave- 
function remains "normalized. That this Slater determinant 
satisfies the Pauli principle follows automatically from the 
property of determinants that interchange of any pair of rows 
or columns reverses its sign. Suppose that we interchange 
electrons 1 and 2, so that electron 1 is put into the orbital 
hitherto occupied by electron 2, and vice versa. The effect on 
the determinant is to interchange the first and second columns, 
and so the sign changes. The same happens when any pair of 
electrons are interchanged, and so the determinant is the 
appropriate combination of the one-electron orbitals. 

It should be observed that the "Pauli exclusion principle 
follows from the disappearance of a determinant when any 
two rows or columns are identical. Suppose that electron 1 
entered orbital \p with spin a and that electron 2 joined it 
with the same spin. Then the first two rows of the determinant 
would be the same, and so it would vanish; therefore it is not 
possible to form a state in which more than one electron 
occupies the same orbital with the same spin. 

A word on notation: the orbitals with their accompanying 
spin are known asspin-orbitafs. A spin-orbital corresponding 
to spin « instead of being written i^* is sometimes written 
merely t/> with the a spin understood. In this notation the 
p 1 spin-orbital is denoted $ . Much paper would be employed 
if a determinantal wavef unction were always written in full; 
therefore it is normally denoted by listing only the terms on 
the diagonal and ignoring (but remembering) ^normalization 
constant. The determinant above becomes l^ a V i^ b - - ■ V z ' 
in this notation. 

It should be noted that only for closed-shell species can the 
wavefunction be represented by a single Slater determinant; 
when the shell is incomplete a linear combination of determin- 
ants must be used. 

Questions. Why is a simple product of orbitals an inadequate 
representation of the state of a many-electron system? Why is 
a Slater determinant a suitable representation? Write the Slater 
determinant for the helium atom, expand it, and confirm by 
inspection that it satisfies the Pauli principle. Do the same for 
four electrons in the lowest-energy configuration of a one- 



dimensional square well. Confirm that the helium atom must 
have paired spins in its ground state, but that in an excited 
state they may be paired (a -singlet) or unpaired (a triplet). 
Repeat these considerations for the hydrogen molecule. 
Continuing with a two-electron system, write down the 
"hamiltonian and show that a simple (non-determinantal) 
product function leads to an expression for the energy in 
which the electronic interactions are represented solely by a 
-Coulombic repulsion term, but that when a Sister determin- 
ant is used an additional integral appears (the -exchange 
integral). What is the value of this exchange integral in the 
ground state of helium? 

Further information. See MQM Chapter 8. See Richards and 
Horsley (1970) for a gentle introduction to the way of manipu- 
lating determinantal wavefunctions, and McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972) for a more 
detailed version. The role of the Pauli principle in determining 
the energies of atoms and molecules is described under 
-Pauli principle, and aspects of the consequences are the 
■exchange energy, Fermi hole, and -spin correlation. The 
•localization of molecular orbitals into regions of a molecule 
can be demonstrated in terms of properties of a Slater 
determinant. 

spectroscopic perturbations. The presence of a 
-perturbation generally muddles a system by shifting energy 
levels and causing states to take on to some extent the 
characteristics of other states. Spectra arise from transitions 
between energy levels, and therefore perturbations appear as 
shifts and changes in intensity of the spectral lines. For 
example, one might be following a series of spectral lines 
forming a regular array on a photographic plate, and then in 
the region of a particular frequency the lines lose their 
regularity and the spectrum seems distorted. This is a 
spectroscopic perturbation and has arisen because several 
states that can be mixed by a perturbation have come close 
together in energy. 

Two classes of perturbation are often distinguished: a 
homogeneous perturbation is an interaction mixing vibrational 
and electronic levels, and in a linear molecule it mixes states 
with the same value of the -quantum number A. fKhetero- 



spherical harmonics 221 



Aeneous perturbation is a rotational -electronic interaction 
(as in A -doubling) and mixes states differing in A by 
±1. 

Further information. See §6.19 of King (1964) and §V.4 of 
Herzberg (1950). See also Herzberg (1966), Kronig (1930), 
and Kovac's (1969). Revise -perturbation theory, 'super- 
position principle, "resonance, and 'predissociation. 

spherical harmonics. The spherical harmonics are a set of 
functions of the angular coordinates Q and (as defined in 
Fig. SS), that satisfy the differential equation A T Yn (0, 0) = 
~~ W + ""^£^(0, 0), where A 1 is the legendrian operator 
(see 'laplacian). These may be expressed as simple poly- 
nomials of sin 8, cos $, sin 0, and cos 0, or as polynomials in 
x, y, and z (Tables 22 and 23), and visualized as the vibrational 
modes of a spherical shell. 



colofifude 




FIG. S8. Spherical polar coordinates. 

Each function is distinguished by the labels £ and m: K may 
take any positive integral value, including zero; and m, for a 
given value of 8, may vary in unit steps from — £ to B. These 
properties of £ and m should be strongly reminiscent of the 
properties of "orbital angular momentum, and it is in fact the 
case that the spherical harmonics are the wavefunctions 
corresponding to states of angular momentum H,m. 

The function V o , o [0,0) is a constant (1/2/ 7r) and therefore 
has the same value at all points on the surface of the sphere: it 
can therefore be depicted by the function drawn in Fig. S9 a, 
where V , is the surface at a constant height above the 
spherical surface. A convenient way of denoting this isotropy 




spherical 

harmonic 





auxiliary 

[uiH_rlcn 





nodes 




FIG. S9. (a) /oo(0, 0), its auxiliary, and a cross-section. |bj V ln W, <b\. 
M Y-x>iO. 0). 

(or spherical symmetry) is to mark off the value of the 
function at a point as a length along the radius of the sphere: 
this gives rise to an auxiliary function, itself a sphere of radius 
y<j.o. which contains all the information about the shape of 
V ,o itself. The connexion between this auxiliary function and 
the representation of a Is 'atomic orbital should be noted 
{see p. 108). There are no angular 'nodes in Y 0l0 . and so it 
corresponds to a wave with zero "orbital angular momentum. 



222 



spherical harmonics 



The function Y ll0 [8, <j>) is proportional to cos and is 
independent of <p. It has extrema at the poles and is zero 
on the equator, and may be represented as in Fig, S9 b. The 
auxiliary function may be constructed in the same way as 
before, and it will be observed that in the northern hemisphere 
the wave has a positive amplitude, and that this is denoted by 
+ in the auxiliary function; and that in the southern hemi- 
sphere the amplitude is negative. The resemblance of the 
auxiliary function to the boundary surface of a p^-orbital 
should be noted. Since there is one angular node the wave 
represents a state of angular momentum with £ = 1. An 
observer looking along the z-axis would not see the equatorial 
node, and so the component of momentum about the z-axis 
is zero (and that, it should be recalled, is the significance of 
the quantum number m being equal to zero). 

The other spherical harmonics corresponding to £ = 1 are 
those with m = 1 and — 1 : both these functions are complex 
(but not complicated) and correspond to waves running round 
the z-axis with their amplitudes predominantly in the 
equatorial zone. Combinations of the two functions may be 
constructed that are real, and correspond to standing waves: 
one combination is Y ul + Y lt _ u which is proportional to 
cos $ (inspect Table 23). This has the same shape as / ,, but 
is orientated along the x-axis. The combination Y il} — Vi,_i, 
which is proportional to sin 0, is also of that shape and is 
directed along the je-axis. The connexion of these combinations 
with the p - and the p -orbitals should be appreciated. 

The procedure may be continued to include the five functions 
with £ = 2, and the functions generated have a close 
resemblance to the d-orbita!s of atomic theory (Fig. S9 c 
and Fig. H12on p. 105). In each case the number of angular 
nodes is equal to £ (we count each plane surface as a node) and 
the connexion of the number of nodes and the angular 
momentum arises as a consequence of the e de Broglie relation 
and the connexion between the "kinetic energy of a state and 
its -wavefunction. The orbital shapes and the atomic-orbital 
boundary surfaces are not identical, because the latter are 
boundaries enclosing chosen amounts of amplitude or density 
and the dependence of these on the radius (as well as the 
angles) distorts the spherical harmonic shapes. 

If it is desired to find the distribution of a particle sliding 
round the surfece of a sphere with a particular angular 



momentum, then in accord with the Born interpretation (see 
-wavef unction) it may be calculated by taking the square of 
the appropriate function. Thus a particle with zero angular 
momentum [S ■ 0) is spread evenly over the surface; one with 
£ = 1 is found either predominantly in the polar regions 
[m = 0) or in the equatorial (m = ±1); one with £ = 2 
is found strongly in the polar regions, but also with a significant 
density spread smoothly around the equator (m = 0), or in 
two bands concentrated at 45° N and 45° S (m = ±1), or 
in a band highly concentrated on the equator [m — ±2). 

Questions. 1. What is the significance of the spherical 
harmonics in classical theory and in quantum theory? What 
values may the indices £ and m take, and how many values of 
m are permitted for a particular value of £? What is the shape 
of V ,o, and why is it plausible that it corresponds to a state of 
zero orbital angular momentum? To what vibration of the 
surface of a sphere does it correspond? Construct the 
auxiliary function for a p^-, a d^.i-, and a d -orbital. Draw 
the shape of the function that determines the probability 
distribution of the particle in each of these orbitals. To what 
vibrations of the sphere do these orbitals correspond? How 
could each vibration be stimulated? Repeat the exercise for 
an f-orbital (£ = 3). Compare your answers with the represen- 
tation of the "hydrogen atomic orbitals depicted in Fig. H12 
on p. 105, Why should the contour diagrams there resemble 
(rather than be identical with) the auxiliary functions you 
have drawn? Suppose you took a spherical shell around the 
proton in the hydrogen atom and at each point plotted the 
value of the wavefunction in the various orbital states, might 
you expect the function so obtained to be identical to the 
functions you have been drawing in this section? Why is that? 
Calculate the latitude of the maximum concentrations of 
electron density in the orbitals d , d^, f . 
2. Confirm that the functions V , o , Vi.i. M, 3 do indeed 
satisfy the differential equation that defines the spherical 
harmonics. Use the legendrian operator in Box 11 on p. 124. 
Apply the operator £ (see Box 14 on p. 161) to the explicit 
expression for the general form of the spherical harmonics 
and confirm that Y^ is an eigenf unction of £ ? with eigen- 
value mti. Prove that the standing- wave solutions V^ m ± 
Y are eigenvalues of the magnitude of the angular 



spin 



223 



momentum, but correspond to states with zero component of 
angular momentum about thez-axis (as should be expected for 
standing waves), and are not eigenf unctions of the operator £ 
(see "quenching). 

Further information. See MQM Chapter 3 for a further dis- 
cussion of the spherical harmonics, their connexion with 
angular momentum, and illustration of auxiliary functions. A 
pleasing account of the properties of the spherical harmonics 
and their relation to states of particular orbital angular 
momentum, to waves on a flooded planet, to tidal motion, 
and to the vibration of spheres is given by Kauzmann (1957). 
See Kauzmann (1957) also for the solution of the differential 
equation for the spherical harmonics, and Pauling and Wilson 
(1935). Alternative derivations are provided by MQM 
Chapters, Rose (1957), Brink and Satcbler (1968), and 
Edmonds (1957). The mathematical properties of the spherical 
harmonics, and their components the associated Legendre 
functions, are given in detail by Abrarnowitz and Stegun 
(1965). 



spin. The spin is the intrinsic, characteristic, and irremovable 
"angular momentum of a particle. A convenient fiction is to 
suppose that the spin is the angular momentum arising from 
the rotation of a body about its own axis: this model enables 
one to recall most of the properties of quantum-mechanical 
spin, and in particular to understand (albeit at only a shallow 
level) why charged particles with spin also possess an intrinsic 
magnetic moment. The quantum-mechanical description of the 
spin of a particle leads to the following conclusions. 

1 . The magnitude of the spin angular momentum of a 
particle is determined by the spin quantum number s which has a 
positive, unique, integral or half-integral value characteristic of 
the particle. If the value of s is known the magnitude of the 
spin can be calculated from the expression \s{s + 1 )] y 'h. As 
an example, the spin of an electron is s = \; this means that its 
spin angular momentum is 3^fi/2, or 0-91 X 10" 34 J s, what- 
ever its other state or condition (the spin is an ineluctable 
characteristic of the particle, like its charge). Each nucleus has 
a characteristic spin, and the letter / is used in place of s as the 
nuclear spin quantum number. See Table 17 for a list of the 
spins of some nuclei. 



2. The orientation of the spin angular momentum is 
quantized (confined to particular angles) in the manner of all 
'angular momenta. The orientation is determined by the value 
of the magnetic quantum number m : m h is the value of the 
component of spin angular momentum on an arbitrary axis in 
space (conventionally thez-axis). As an example, two values 
of m are permitted for the electron (s = h, namely m =*+«; 
corresponding to a component of magnitude ~li on the z-axis, 
and m — — ^, corresponding to a component — ^fi on the 
z-axis. The different signs are often referred to as denoting an 
'up-spin' (or a-spin) or a 'down-spin' (or /3-spin), and the good 
sense of this can be appreciated from a "vector model of the 
situation. 

3. Spin is a non-classical phenomenon in the sense that if 

ti were zero the spin angular momentum would vanish. Do not 
draw the conclusion that all angular momenta are non-classical 
in the same sense: an orbital angular momentum of magnitude 
[£(£ + 1)3 fi could survive the dwindling of li to zero in a 
classical world because fi can be increased without limit so that 
the product did not vanish; but the value of s is fixed. 

4. Spin is not a relativistic phenomenon in the sense 
normally put on these words; thus although spin emerges 
naturally as a consequence of 'Dirac's relativistic equation it 
is possible to arrive at its existence without referring to 
relativity. 

5. Spin is a fundamental classifier and divides all matter 
into two camps with fundamentally different behaviour. 
Particles with half-integral spin are called 'fermions and 
satisfy Fermi-Dirac statistics; particles with integral spin 
(including zero) are -bosons and satisfy Bose-Einstein 
Statistics. The two classes satisfy different forms of the "Pauli 
principle, and because of this they show profoundly different 
behaviour. It Is just possible that there exists a third camp 
containing the parafermions which are neither fermions nor 
bosons and might be needed to account for the properties of 
quarks, if these exist. 

6. The tendency of spins to •pair is a term too often ill-used 
in fallacious accounts of chemical bonding, where it is quoted 
as the reason why bonds form. Energy considerations govern 
bonding, and if by pairing electrons are enabled to enter a 
low-lying orbital, and so reduce the energy of the molecular 
system below that of the separated atoms, then pairing will 



224 



spin-correlation 



occur. But rather than showing any transcendental mutual 
affection they are forced to pair (essentially by the "Pauli 
exclusion principle) in order to achieve this low-energy state. 

Questions. What is spin? What do the quantum numbers s and 
m signify? What angle does the spin-momentum vector make 
to the z-axis in the a-spin state of an electron? What is the 
minimum angular momentum of a -photon? Why is spin non- 
classical? Which of the f ol I owi ng species are ferm ions and wh ich 
are bosons: e, p, n, 4 He, 3 He, 2 H + , photon, H 2 , and quark? Why 
do electrons seem to show a tendency to pair? 

Further information. See MQM Chapter 6 for more infor- 
mation, especially information about coupling spins together. 
See Dirac (1958) for a horse's mouth account of spin, and 
Salem and Wigner (1972) for speculations on parafermions. For 
the anguiar momentum of spin see "angular momentum and 
references therein. For an account of the manifestations of 
spin see Wheatley (1970), and McWeeny (1970) for a more 
sophisticated version, For an account of other properties that 
behave like spin (for example, charge) see Lipkin (1965) and 
Lichtenberg (1970). Spin was introduced empirically by 
Uhlenbeck and Goudsmit {1925, 1926) in order to explain 
features of 'atomic spectra, developed into a consistent theory 
by Pauli, and then shown to be a consequence of "Dirac's 
equation. The historical development of the idea is described 



in §3.4 of Jammer (1966), a book well worth turning to in 
order to see the emergence (and sometimes eclipse) of the 
unconventional. See also "Stern-Gerlach experiment for an 
earlier experiment proving the existence of spin, but not 
interpreted then as such. Why spin is not a necessarily 
relativistic phenomenon is described by Gaiindo and 
Sanchez del Rio (1961). 

Spin-correlation. Electrons with parallel "spins tend to stay 
apart, and those with opposite spins tend to bunch together. 
This remarkable phenomenon has nothing to do with the 
charge of the electron (although it affects the average 
Coulombic repulsion of two electrons and appears in the 
"exchange energy); nor is it, one presumes, supernatural. The 
tendency is an intrinsic property of electrons and is a con- 
sequence of the "Pauli principle. This may be seen by con- 
sidering a wavef unction for two particles \p{ri, r 2 ), and 
supposing that the particles have no interactions. Then the 
wavefunction can be written as the product ^ a (r[ li^bta'- 
where particle 1 occupies orbital a and particle 2 occupies 
orbital b. If such a wavefunction is to accord with the Pauli 
principle it must be modified to ^ a (ri)i// b (r 2 ) + i// 8 (r 2 ) ]/> b (r ( ) 
if the spins are paired, and to 4> a (*\)i> b bi) ~ & l (ra)lM lr l) if 
they are parallel. In the latter case we can investigate the 
probability of finding both electrons at the same point by 



FIG. S10. Formation of 8 Fermi 
hole. Contours of total wave- 
function for two non-interacting 
particles in a one-dimensional 
square well. Note that node at 
x i = X2 when the spins are 
parallel. 





Porollel spins 



Paired CantiporollelJ spins 



spin-orbit coupling 



225 



letting r, and r 2 coincide: the wavefunction becomes 
<MM>Mi'i)~!M r ]l | M r i). and vanishes. Therefore we 
must conclude that there is a vanishing probability of finding 
both electrons at the same point if their spins are parallel. 
In Fig. S10 is illustrated the situation for two particles 
(electrons, with their charge ignored) in a one-dimensional 
square well (a narrow wire). The contours depict the prob- 
abilities of finding the particles at points x, and x 2 in the wire 
when one particle occupies the ground state (n = 1, see 
•particle in a square well) and the other occupies the first 
excited level [n = 2). The mutual avoidance when the spins are 
parallel (Fig. S10 a) is manifest as the disappearing probability 
along the line x t = x 2 . If one imagines an electron at some 
point, then one may imagine a small surrounding volume into 
which electrons with the same spin direction will tend not to 
penetrate: this region is referred to as a Fermi hole. An 
electron with a spin opposite to the first is described by the 
other wavefunction, and will have an enhanced probability 
of occurring within this small volume; although the resulting 
bunching of electrons is not, as far as I know, referred to as a 
Fermi heap. Thus we see that the relative positions of electrons 
tend to be correlated by virtue of their relative spins. 

Further information. See MQM Chapter 8 for a further dis- 
cussion and another picture. There is a brief account of the 
problem in §IV.6 of Linnett (1960), who gives pictures like 
Fig. S10 but for the helium atom: these are instructive. 
Linnett (1964) has emphasized spin-correlation effects in a 
theory of chemical bonding. Reviews of electron correlation in 
atoms and molecules include those by Paunch (1969) and 
Hylleraas (1964), See "exchange energy and -self -consistent 
fields for more information. 

Spin-orbit coupling. An electron in an atom possesses a 
magnetic moment both by virtue of its "spin and its "orbital 
magnetic moments. Two magnetic moments in each other's 
vicinity interact, and the strength of the interaction depends 
on the magnitudes of the moments and their mutual orien- 
tation: this interaction energy appears in atomic spectra as 
"fine structure. 

The source of the spin-orbit coupling energy can be visual- 
ized by taking up a position on an electron which is orbiting a 
nucleus of charge Ze. Such an observer will see a positive 



current encircling him, and a consequence of the current is a 
magnetic moment at hts position. Thus we can conclude that 
the electron spin magnetic moment is bathed in a magnetic 
field arising from its own orbital motion. (There will also be 
magnetic fields arising at the electron by virtue of the orbital 
moments of the other electrons in the atom or molecule, but 
these spin-other-orbit interactions are generally less important.) 
It Is in principle a simple matter to calculate the magnetic 
field B at the electron by virtue of its orbital motion, and 
therefore to deduce its magnetic energy from the expression 
"9% S - B (because the interaction energy is — m.B, and the 
spin "magnetic moment \sgy^); then we could anticipate that 
B is proportional to I, the orbital angular momentum, and so 
expect an energy of the form (f/h 2 )s.l, where f is the spin- 
orbit coupling constant Nevertheless, the calculation is not 
quite so straightforward, and the early, straightforward 
physicists, who first did it, were perplexed when they obtained 
disagreement with experiment. 

The calculation proceeds by finding the magnetic field at 
a body moving in a electric field with a particular velocity. In 
this case the electric field is due to the nuclear Coulomb 
potential, and the velocity may be expressed in terms of the 
orbital angular momentum through 1 = r A mv. Then the 
resulting radius-dependent quantity is averaged over the radial 
part of the atomic orbital occupied by the electron, and the 
mean constant so obtained is identified with JYh 2 . This 
approach gives an answer which is broadly correct (f is 
strongly dependent on Z, and varies as Z 4 ) but is wrong by a 
factor of 2. With "spin we know that factors of 2 are not 
unreasonable; but even so they must be justified. One 
justification for this extra factor may be found from the 
"Dirac equation; from this the correct expression emerges. 
Nevertheless, it is always pleasing to seek a more physical, 
visualizable explanation, and as usual one may be provided. 
When we step on to the orbiting electron in order to observe 
the current due to the central nucleus, we must do so with 
more circumspection than had Copernicus when he trod the 
opposite journey. In the atom the electron is moving so fast 
that it must be regarded relativistically, and watching an 
electron spin from the viewpoint of a nucleus is not the same 
as watching it spin from the viewpoint of an observer travelling 
with it. By a coincidence, or by design, the electron is moving 



spin-spin coupling 



in such a way that to an observer on the nucleus it appears to 
be spinning at only one-half its rate for a travelling observer. 
This modification of its motion, which is essentially Thomas 
precession, introduces an extra factor ^ into the spin-orbit 
calculation, and so brings it into conformity with the "Dirac 
equation, and with experiment. 

The strength of the spin-orbit coupling constant increases 
with the atomic number of the atom; the heavy atoms have 
large spin-orbit coupling constants (some are listed in Table 9 
on p. 271). In one-electron atoms this dependence isZ 4 as 
mentioned above, and this reflects the dominant sampling by 
the electrons of regions close to the nucleus where the field is 
strong: as Z increases the orbital contracts and the electric 
increases, and both lead to a larger value of J. Outer electrons 
are further away, and in hydrogen-like atoms J falls as 1/n 3 : 
this also reflects the magnitude of the electric field sampled by 
the electron. 

Further information. See MQM Chapter 8 for a detailed 
examination of the topics mentioned here, including the 
calculation of f for hydrogen-like atoms, and a further 
mention of Thomas precession. The latter is well-described in 
Moss (1973) and Hameka (1965). The role of shielding 
electrons is examined further in MQM. A thorough discussion 
of spin-orbit coupling effects, especially with respect to 
"singlet and triplet states and "phosphorescence, will be 
found in McGIynn, Azumi, and Kinoshita (1969): they 
explain how to do calculations involving spin-orbit coupling, 
and tabulate data. For applications in atomic spectroscopy, 
see "fine structure and its ramifications. See Table 9 for some 
spin-orbit coupling data. 

spin-Spin coupling. The interaction between nuclei that in 
"nuclear magnetic resonance (n.m.r.) gives rise to the splitting 
known as fine structure is called spin-spin coupling, and its 
magnitude is generally denoted J and quoted in hertz {Hz, 
cycles per second}. The appearance of a spectrum showing fine 
structure is shown in Fig. N3 on p. 154. The coupling between 
most protons lies in the range —20 Hz to +40 Hz, although 
the commonly observed range is from Hz up to 10 Hz. These 
energies are minute, and correspond to a magnetic field being 
induced at one nucleus by virtue of the presence and orien- 
tation of another. 



One possible mechanism for the interaction is a direct 
"dipole-dipole interaction between the two nuclear "spin 
"magnetic moments; but the spherical average of such an inter- 
action is zero, and so it cannot contribute to the fine structure 
in a molecule freely rotating in a fluid medium. 

Another interaction involves the electrons in the bonds as 
intermediaries in transmitting the interaction between the 
nuclei. This mechanism may be illustrated by the example of 
the hydrogen molecule: the problem is to account for the fact 
that the energy of the molecule with the nuclear spins parallel 
is different from that of the molecule with the nuclear spins op- 
posed. The key to the mechanism is the "hyperf ine interaction 
between the nuclei and the electrons. Consider the case when 
an a electron is close to nucleus A; by virtue of the "Fermi 
contact interaction the electron and nuclear magnetic 
moments couple, and the lower-energy configuration is with 
antiparallel spins {parallel moments) {Fig. S1 1). But the 
"Pauli principle requires the other electron in the bond to be 
antiparallel to the first, and charge correlation suggests that 
the other electron, with (3 spin, will be predominantly in the 
vicinity of the other nucleus. With that nucleus the second 
electron has a hyperf ine interaction by virtue of the Fermi 
contact term, and so the lowest-energy configuration for the 
other nucleus is with 0spin. Therefore, overall we see that if 
one nucleus has a particular spin then the other attains the 
lowest energy if it has the opposite spin. It follows that the 
energy of the molecule with parallel nuclear spins lies very 
slightly above that with opposed spins; consequently it 
requires energy to turn the spins into a parallel alignment. 




FIG. S1 1. Spin-spin coupling between protons. 



The magnitude of the effect depends on the strength of the 
hyperf ine interactions involved, and on the ease with which 
the electrons In the bond can be polarized to wash predomi- 
nantly over the most favourable nuclei: the latter depends on 
the mean excitation energy of the bond, and since the 
tendency for the electron spins to be decoupled must be 
taken into account, it turns out that the mean energy required 
is the mean energy of excitation to triplet configurations of 
the bond (see "singlet and triplet states). The interaction does 
not average to zero as the molecule rotates because the contact 
interactions are isotropic and do not themselves average to 
zero. 

No account of the external magnetic field was taken in the 
mechanism, and therefore it should be expected, and is indeed 
found, that the spin-spin coupling interaction is independent 
of the strength of the applied field. 

We have shown that the interaction energy depends on the 
relative orientation of the two nuclear spins; therefore it is not 
unreasonable to expect an interaction proportional to the 
scalar product i A .l 8 : this form of the interaction is found by 
detailed calculation, and the constant of proportionality is just 
the spin-spin coupling constant JPh. (If J is positive, the anti- 
parallel orientation of the spins is the more stable because 
W&H A .I B then gives a negative contribution to the total 
energy.) 

The example we have provided is artificial in one sense: the 
spin-spin coupling can be detected in an n.m.r. spectrum only 
when the coupled nuclei are in different chemical environ- 
ments (have different "chemical shifts). Thus although there 
is a coupling between the protons in H 2 (with a magnitude of 
280 Hz), the n.m.r. spectrum consists of a single line because 
all the allowed transitions occur at the same energy (see 
•n.m.r.). It is more realistic to consider proton coupling in 
more complicated molecules, and the mechanism already 
described may be extended in a simple fashion by considering 
the chain of interactions through the bonds as well as the 
possibility of direct interaction by the overlap brought about 
by squashing two nonbonded atoms together. 

In an obvious notation, using large and small arrows to 
denote electron and proton spins respectively, the interaction 
in H z can be represented by j t I j \ } for the lower energy 



spin-spin coupling 227 

orientations, and j t I | t | for the upper energy. Analog- 
ously, in the CH 2 group the chain will be j f I I f [ 
where the parallel orientation of the middle electron pair is 
favoured by the "Hund rule of maximum multiplicity for 
atoms (which favours parallel arrangements of spins on atoms); 
the upper energy orientation for this group is 

\ M i ! i ] J I • and '* ' s i m P ortant to note that, 
because of the insertion of the atom in the chain of inter- 
actions, the parallel nuclear orientation lies below the anti- 
parallel (corresponding to a negative./). This chain of 
interactions can be quite large: J can lie in the range 
— 21 -5 Hz < J < 42-4 Hz. An example is the coupling in HCHO 
(42'4 Hz). A coupling between two protons separated by three 

bonds proceeds through the chain j f \ t \ 1 1 Mt|| 

and leads to an antiparallel arrangement as the low-energy 
state U positive). Beyond three bonds the interaction is 
strongly attenuated -that is, much reduced-this is fortunate, 
for otherwise n.m.r. spectra would be impossibly complicated 
to disentangle. 

The spin-spin coupling between protons separated by bonds 
depends on the conformation of the molecule. This is because 
the efficiency of alignment of the spins at the intervening 
atoms depends on the relative orientation of the bonds. Thus 

the predominance of |—{it ■■■■{ over {•••♦fii—*} 
depends on factors like the -hybridization of the atoms. As an 
example, the trans coupling in ethene is 19-1 Hz whereas the 
ci's coupling is 11-6 Hz. 

Although we have dismissed the direct dipole-dipole inter- 
action in fluids, the electron-nucleus dipolar hyperf ine 
interaction can contribute to spin-spin coupling. We have 
seen that the spin-transmission mechanism involves two 
hyperf ine interactions, one at each end of the chain; the 
rotational average of the product of two dipolar interactions 
does not disappear, and so two electron-nuclear dipole inter- 
actions, one at each proton, can contribute. This type of 
interaction is important in atoms other than hydrogen where 
p-orbitals occur in the valence shell. 

Questions. What is the significance of the term 'spin-spin 
coupling' in n.m.r.? Under what circumstances will a 



Stark effect 



coupling not show in the spectrum even though it is non-zero? 
What is the range of proton coupling constants in n.m.r.? 
What dependence on the strength of the applied field do they 
show? Why is the direct dipole-dipole interaction between two 
spins unimportant in fluid media? Is it important in solids? 
What is a typical magnitude for the direct dipolar interaction 
between the two protons in the water molecule? Investigate 
how this interaction depends on the orientation of the mole- 
cule with respect to the applied field {assume that the proton 
moments align themselves with respect to this field and search 
Box 5, p, 50, for formulae). What structural information 
might you anticipate obtaining from a study of the direct 
interaction? What is the source of the spin-spin interaction 
between protons in liquids? Explain the sequence of inter- 
actions that transmits the orientation of the proton to its 
neighbour. What happens to the interaction when the protons 
are separated by 3, 4, and 5 bonds? What excited states should 
be mixed into the ground state in order to yield the spin 
polarization of the bond? Can a dipolar interaction give any 
contribution in fluids? What is the significance of a negative 
value of J? 

Further information. See MQM Chapter 11 for a detailed dis- 
cussion of the source of the spin-spin coupling. The role it 
plays in n.m.r. is described by McLauchlan in Magnetic 
resonance (OCS 1). Further details will be found in Lynden- 
Bell and Harris (1169), Carrington and Mctachlan (1967), 
Slichter (1963), Memory (1968), and Abragam (1961). The 
analysis of n.m.r, spectra in order to extract coupling data, 
and its application, are described by these books and in 
Roberts (1961), Abraham (1971), Emstey, Feeney, and 
Sutdiffe (1965), Pople, Schneider, and Bernstein (1959), and 
Corio (1966), The reason why the spectrum exhibits coupling 
between non-equivaient nuclei can be seen by referring to the 
little piece of mathematics in §4.4 of Carrington and 
McLachlan (1967). Tables of J values will be found in Emsley, 
Feeney, and Sutdiffe (1965). 

Stark effect. The modification of the energy levels, and 
therefore of the spectra, of atoms and molecules by the appli- 
cation of an electric field, is known as the Stark effect. It is 
possible to distinguish the first- and second-order effects 



(which are respectively linear and quadratic in the strength of 
the applied field) and the atomic and molecular effects. 

The first-order atomic Stark effect is large but rare, for it 
depends on the presence of a -degeneracy which enables the 
atom to respond massively to the applied field. Such a 
situation occurs in atomic hydrogen: because the 2s- and 2p- 
orbitals are degenerate, when a field is applied the electron 
can easily reorganize itself by -hybridization. Thus the com- 
bination tp 2s + i*%p S' ves a distribution strongly biased 
towards the low-potential region, and the other possible 
combination i/^s — lK p is concentrated on the high-potential 
side of the nucleus (Fig. S 12). Therefore the transitions 
involving the n = 2 shell of the atom arc no longer degenerate 
and occur at different frequencies. The p - and p -orbitals 

x y 

are unaffected by the field. 




field off 



field on 



FIG. S12. Linear Stark effect in atomic hydrogen. 

As in the -Zeeman effect the transitions are polarized. Two 
points follow from our discussion of -electric dipole 
transitions. 

1. If m% does not change, the emitted light (forming the 
ir-lines) is polarized parallel to the direction of the applied 
field (and so it would not be seen if viewed along the direction 



Stark effect 



229 



of the field; it is radiated in a belt around the transverse 
direction). 

2. If m e changes by +1 the light (the a-llnes) is transversely 
polarized. When viewed perpendicular to the field the 
Amj; = +1 component is right circularly polarized and the — 1 
component is left circularly polarized. When viewed along 
the field direction the light is unpolarized because the +1 and 
—1 transitions occur at the same frequency, and so the light 
emitted from different atoms gives an incoherent superposition 
of both circular polarization states. 

Not only does the Stark effect cause a splitting of lines, it 
also causes them to broaden slightly, and shifts the series 
limits to lower frequency. Both effects are related to the presence 
of the low potential on one side of the atom, for it enables an 
electron to escape (Fig. S13). An electron in a state not far 
from the ionization limit may be able to -tunnel through the 
remaining potential -energy barrier and emerge into a region 
where the applied field can pluck it from the atom. This 
reduces its lifetime in the excited state, and so, by the 
"uncertainty principle, the energy of that state is made 
imprecise; this imprecision appears in the spectrum as a 
broadening of the appropriate lines. The electron need not 
be excited into so high an energy state for ionization to occur, 
and so the field also reduces the energy of the series limit 
(Fig. S13). 




appoint J.R- 



FIG. S13. Extra consequences of the Stark effect for hydrogen. 



In atoms not having the degeneracies possessed by hydrogen, 
the first-order effect is absent and is replaced by the much 
weaker second-order effect : One power of the field is used 
in distorting the atom from spherical symmetry ("polarizing it), 
and the second power is used in the interaction with the dipole 
moment of the distorted atom, so causing the energy 
separation. Since normal fields can polarize atoms only slightly 
the induced dipole is small and its interaction with the field 
weak; therefore the second-order Stark energy shifts are small 
and huge fields (~ 10 s V cm" 1 ) must be used. 

The molecular Stark effect may also be of first- or second- 
order. The first-order effect is shown by symmetric top 
molecules with permanent dipole moments: the applied field 
causes "rotational states of the same value of J but different 
values of M to have different energies (they are degenerate in 
the absence of the field), and the splitting is proportional to 
the permanent dipole moment of the molecule (i, as we 
explain below. Since fields of the order of 50 kV cm" 1 give 
splittings of the order of 20 MHz, and this is easily and 
accurately detectable in a microwave spectrum, the method is 
a powerful way of determining dipole moments. 

It is instructive and quite easy to understand why the 
energy of a state with quantum numbers J, K, and M is 
shifted by an amount -pMKBUKJ + 1 ) by a field E , From the 
theory of the "rotation of symmetric -top molecules we know 
that K is the projection of the -angular momentum J on the 
figure axis, and that J "precesses around this axis. An alterna- 
tive view would be to consider the vector J as fixed and the 
axis as precessing around it. Since the dipole is directed along 
the axis the motion averages out its components except for 
one of magnitude jncosO along J (see Fig. SI 4), But J is also 
precessing about the field direction, and the component of 
dipole parallel to the field isjucosfl cost 1 '; therefore the inter- 
action energy is — jufcosfl cosf?'. From Fig. SI 4 we can use 
simple trigonometry to express cost 1 as Kl[J(J + 1)) % and 
cos8' as M/UV + 1 )] Vl . Combining these results we obtain 
the energy as —yMKEUV + 1 ), as we anticipated. The appli- 
cation of fiie "selection rules &J= ±1, AK = 0, and AAf = 0, 
±1 enables the spectrum (and from the M changes, the 
polarization of the lines) to be predicted. 

In the case of linear molecules, where the angular 
momentum is perpendicular to the dipole moment, and in 



230 



Stern-Gerlach experiment 




FIG. S14. Molecular Stark effect. 

asymmetric tops, where the lack of symmetry causes the 
motion to be very complicated, there is no linear effect, but 
the orientating effect of the field (which, in classical terms, 
distorts the rotational path of the molecule) affects the 
energy of the states and induces energy shifts of the form 
IJ 1 E 2 {A 1 + A t M 2 ). 

The molecular Stark effect is of considerable importance in 
the study of permanent dipole moments of the molecules that 
can be examined by microwave spectroscopy, and it is also 
important in the technology of microwave spectrometers, for 
an oscillating electric field (usually of several 10 kV cm" 1 ) will 
modulate the position of spectral lines, and therefore the 
intensity of absorption or emission at a particular frequency. 
Detectors making use of this oscillation of the intensity are 
called Stark-modulation spectrometers. 

Questions. 1 . What is the Stark effect? What classifications of 
the effect are there? Why is the first-order atomic effect con- 
fined to hydrogen-like atoms? Would you expect it to be 
shown by helium in a highly excited state? What polarizations 
are observed for the permitted transitions in hydrogen? Take 
the hydrogen energy-level diagram (Fig. G3 on p. 86), modify 
it to show the effect of an electric field, and discuss the form 
of the spectrum. The shift of the energy of an electron in the 



n = 2 shell results in a separation of the levels by an amount 
Qea a E, where a is the "Bohr radius. Compute the separation 
for an applied field of 20 kV cm" 1 and 100 kV cm' 1 . What 
separation would be expected for He + in the same state and in 
the same fields? What other effect does a strong field have? 
What effect has the field in the case of atoms that lack 
hydrogen-like degeneracy? Why is the second-order effect 
small? What effects occur in molecules? The permanent dipole 
moment of ammonia is T47 D; discuss the splitting of the 
J = 0, 1 , 2 states brought about by a field of 50 kV cm" 1 ; 
sketch the form of the spectrum and label the polarizations of 
the lines (ignore, if you are aware of them, the effects of 
•nuclear statistics). 

2. Using the explicit orbitais given in Table 15 on p. 275 set up 
the 2 X 2 secular equation for the effect of an electric field 
along z on the energies of the 2s- and 2p-orbitals of hydrogen. 
Solve for the energies and states, and evaluate all integrals. 
The splitting of the 2s- and 2p-orbitals is, as stated in the 
proceeding question, 6ea E, and this should be your answer. 

Further information. See MQM Chapter 8. A nice, and 
occasionally anthropomorphic, discussion of the effect on 
atomic spectra is given in Chapter 17 of Condon and Shortley 
(1963); see also §111 A4 of Kuhn (1962), Chapter 20 of 
White (1934), and §11.3 of Herzberg (1944). The molecular 
effect, together with accounts of how to determine dipole 
moments and build spectrometers using Stark modulation, is 
described in Chapter 7 of Sugden and Kenney (1965) and in 
Chapter 10 of Townes and Schawlow (1955). 

Stern-Gerlach experiment. The Stern-Gerlach experiment 
consisted of the passage of a collimated beam of atoms (silver 
atoms boiled off hot metal through slits into a vacuum) 
through an inhomogeneous field, and the observation of the 
distribution of the atoms when the beam had been deposited 
on a glass plate. If the atoms possessed a magnetic moment 
the effect of the inhomogeneous field would be to drive in one 
direction those that had one orientation, and in the opposite 
direction those that had an opposite orientation, and, accord- 
ing to classical physics, to spread atoms with all intermediate 
orientations into the region in between. An inhomogeneous 
magnetic field is essential for this effect because a homogen- 



eous field would not split the beam as it provides no 
directional information; an inhomogeneous field, one with a 
non-vanishing gradient, provides a sense of direction. 

In their first experiment Stern and Gerlach observed that 
classically anticipated result. In their second, done with great 
care with a low pressure and a long exposure, they saw that 
the band of deposited atoms had two closely spaced com- 
ponents, separated by a clear region. This result is wholly at 
variance with classical physics but in full accord with 
quantum theory, for each silver atom possesses an unpaired 
electron with "spin, and therefore has a 'magnetic moment. 
Quantum theory predicts that a spin-j object can take only 
two orientations in a magnetic field, and so the Stern- 
Gerlach experiment confirms this in a striking fashion. 

It is important to note that this was the first piece of 
evidence for the quantum theory that did not involve a 
thermal experiment or an experiment involving radiation: it 
provided a purely mechanical demonstration of quantization 
{space quantization, as the restricted number of orientations 
of an angular momentum is termed). The original explanation 
did not associate the magnetic moment with the intrinsic -spin 
of the electron; that came later (1925) when Uhlenbeck and 
Goudsmit introduced the concept from their study of atomic 
spectra. The experiment was also one of the first applications 
of molecular beams, a subject now in a dynamic phase. 

Questions. Sketch the Stem-Gerlach experimental arrangement. 
Why is it necessary to use a very low pressure in the apparatus? 
Why is it necessary to use an inhomogeneous field? What is 
the result predicted by ctassical physics for the experiment? 
What is the result obtained? Why is the result consistent with 
quantum theory? Suppose the upper beam were passed through 
another inhomogeneous field with (a) the field in the same 
direction as the first, and (b) the field rotated about the 
direction of the beam by 90°: what would be the result 
of the experiment? 

Further information. The Stern-Gerlach experiment is put 
into its historical context in §3.4 of Jammer (1966). For the 
details, see §134 of Richtmeyr, Kennard, and Lauristen (1955) 
and §V1,2 of Ramsey U956). The original papers are by Stern 
(1921) and Stern and Gerlach (1922). An analysis wilt be 
found in §14.16 of Bohm (1951). For the philosophical dis- 



superposition principle 231 

cussion to which the result gave rise, see Einstein and 
Ehrenfest (1922) and Jammer (1966). The modern state of 
molecular beams is described in Ross (1966) and Ramsey 
(1956); see also Levine and Bernstein (1974). 

superposition principle. The superposition principle states 
that when a situation is a composition of a number of 
elementary situations, its amplitude is the superposition of the 
amplitudes for the components. The significance of this 
principle, which is one of the fundamental principles of 
quantum mechanics and implies the notable features of the 
differences between classical mechanics and quantum 
mechanics, can be introduced by considering the classical 
situation. 

Suppose that an event can be divided into a number of 
composite events; for example, the event might be the journey 
from a point p x to another p 2 , and the elementary events 
might be the journeys by alternative paths through the points 
P a or P b - Classical theory ascribes a probability P{a) to the 
path through p a and a probability P(b) to the path through p , 
and goes on to say that the total probability of making the 
journey from p, to p 2 is the sum of the probabilities of making 
the individual journeys: P = P(a) + P(b) (Fig. SI 5). 



Classical 




PCa) 




PCM 



(/ —+0 

FIG. S15. The superposition principle. 



PCoHPCW 



Quanta! 



Pidi=f\o)fto) 



pca=Y*tw^cb) 



l^faj+^cal" 



232 



symmetry operation 



This discussion might seem trivial; but quantum theory 
shows that it is wrong. The superposition principle agrees that 
there is a probability of effecting the journey through p with 
a probability P{a), and an analogous probability for the path 
through p b , but it disputes the assertion that the joint prob- 
ability is the sum of the probabilities. It implies that in order 
to compute the joint probability it is first necessary to find 
the probability amplitude (the °wavef unction ) for the 
individual paths, then to find the probability amplitude for 
the composite situation, and from that to find the probability 
itself (Fig. S15). Let the probability amplitude for the journey 
through p be i^(a): by this we mean that the probability itself 
is ^*(a}i^{a). The corresponding probability amplitude for the 
journey through p is !^(b). The superposition principle tells 
us that the amplitude \p for the composite process is the sum 
<^(a) + \jj{b). It follows that the probability for the joint 
process is $*$), which may be expanded into P(a) + P(b) + 
^*{a)^ib) + i^(a)^*{b}. This total probability differs from 
the sum of the individual probabilities by the interference 
term ip*{a)ip(b) + 0{a)i//*<b). 

First we should note that if one of the paths were forbidden 
its amplitude would be zero and the probability for the overall 
journey would reduce to P = P(a) (for p b closed). Next we 
should note that the interference term does not occur in the 
classical case, is characteristic of quantum processes, and arises 
from the insistence of the superposition principle that ail basic 
manipulations are done on the amplitudes. 

If the superposition principle is a true description of 
Nature we should be able to detect some direct physical con- 
sequences: one that might immediately spring to mind is the 
possibility of observing interference effects for particles that 
are able to travel from a gun at p t to a target at p 2 through a 
screen perforated with two holes: such interference effects of 
particles, which correspond to the analogous interference of 
light under similar circumstances, have been observed. 

The superposition principle applies to all the processes of 
quantum mechanics. Wherever a compound situation is under 
consideration the calculations must be performed on the wave- 
function (the state amplitude): that is why the "Schrodinger 
equation is a central formula of quantum theory— it enables 
the amplitude of a state to be calculated, and shows how to 
determine its time evolution. 



The -LCAO method is an example of the working of the 
superposition principle. To see this, we can take the view that 
the probability of finding an electron at a particular point of 
the molecule depends on its probability of being in some of 
the atomic orbitals of the constituent atoms. Therefore to work 
out the probability of being at the point r we must know the prob- 
ability amplitude for the electron to be there if it occupied 
each of the atoms separately. These probability amplitudes 
are just the atomic-orbital wavef unctions themselves, each 
evaluated at the point r. Therefore the total probability 
amplitude of the electron being at r is a sum of the atomic 
orbitals (at that point), and the probability is the square of 
this amplitude. This is precisely the interpretation of 'mole- 
cular orbitals in terms of the LCAO approximation. 

Questions. State the superposition principle. Discuss the 
process of travelling from Aix to Ghent through two gated 
roads, and take into account both classically and quantum 
mechanically the chance that the gates might be closed. 
Discuss the analogy between the propagation of particles 
and of light in terms of the superposition principle. Discuss 
the diffraction of particles in terms of the superposition 
principle. Why is the LCAO account of molecular structure 
plausible in terms of the superposition principle? Discuss, 
in its terms, the concept of "hybridization. 

Further information. See Feynman, Leighton, and Sands 
(1963), §9.2 of Bohm (1951 ], and Chapter 1 of Dirac (1958). 
. For a simple account of optical analogies see MQM Chapter 2. 
See Jammer (1966) for historical attitudes. See Feynman and 
Hibbs (1965) for a construction of quantum mechanics in 
terms of the superposition principle. 

symmetry operation. A symmetry operation or symmetry 
transformation is a transformation of the coordinates of the 
system {passive convention) or transformation of the object 
itself {active convention) that, after its application, leaves the 
system in a configuration indistinguishable from its original. 
The active convention is the more direct, and we illustrate 
it first. Consider an undecorated square object lying on a plane 
on which is drawn a coordinate system; close your eyes, rotate 
the object through 90° in the plane, and look at it again. It is 
impossible to tell that it has been rotated, and so this operation 



symmetry operation 



233 




FIG. SI 6. (a) Active and (b) 
passive conventions for 

transformations. 



is a symmetry operation. Had the rotation been through an 
angle less than 90° we would have noticed a new orientation. 
Similarly a rectangular (oblong) object can be rotated through 
180 into an indistinguishable position, which rotation is 
therefore a symmetry operation; but rotation through 90° is 
not a symmetry operation (Fig. S16), 



In the passive convention a coordinate system is drawn on 
the plane on which the square object is lying; if we move a 
units along x we arrive at the edge of the square (of side 2a), 
and if we step a units along y we also come to an edge. Now 
rotate the coordinates (that is, the underlying plane rather 
than the object) until thex-axis lies along the direction 



FIG. S17. The symmetry 
operations C , o , i, and S and 
an example of a molecule 
possessing each. 




234 



sy m met ry ope ration 



originally occupied by the y-axis. Stepping along the new 
x-axis a units, we again encounter an edge; likewise for the 
new y-axis. This system is indistinguishable from the first, and 
so the 90 rotation is a symmetry operation. In the case of a 
rectangle of sides 2a and 2b stepping along the original x-axis 
a units will bring us to an edge, but if we step a units along the 
rotated x-axis we shall not be at an edge unless a = b (Fig. 
SI 6). This is therefore not a symmetric situation, and so a 90° 
rotation of a rectangle is not a symmetry operation. 

It is a matter of taste which convention one employs, 
although there are aesthetic advantages (at least to my muse} 
in the passive convention on the grounds that it is always 
possible to manipulate coordinates (they being a convenient 
mathematical fiction, and at our disposal), whereas it may not 
be possible to do the active act. This is especially true in the 
case of symmetry operations other than rotations: inversion 
through a point is difficult to effect actively (although it may, 
of course, be imagined), and so too is reflection. 

The basic symmetry operations are those of translation 
(especially in periodic systems such as crystals or in free space), 
rotation through 2ir/n rad (denoted C ), reflection in a plane 
(a)— a subscript v implying a vertical plane and an/i implying 
a horizontal plane— inversion through a point (i), and rotary- 
reflection (S n ), which is a composite motion effected by 



rotating by 2itln rad and then reflecting in the plane to which 
the rotation axis is perpendicular (see Fig. S17). 

The study of the effect of symmetry operations is the 
basis of "group theory. 

Questions. 1 . What is a symmetry operation? What two con- 
ventions are employed? Discuss in the active and the passive 
conventions the operations denoted C 2 , C 3 , (J, i, S*. Enumerate 
the symmetry operations of the following objects: a left 
hand, a right hand, a man, a chair, a cube, a triangle, a 
tetrahedron, a square pyramid, the molecules H 2 0, C0 2 , CH 4 , 
SF 6 , benzene. 

2. Demonstrate that an alternative and consistent definition of 
a symmetry operation is that it is an "operation that 
■^commutes with the system's "hamiltonian. 

Further information. See MQM Chapter 5 for a further dis- 
cussion of symmetry operations and the development of the 
structure of "group theory. See Atkins, Child, and Phillips 
(1970) for a simple enumeration of symmetry elements and 
some examples. Refer to the section on "group theory for 
further references, and see Weyl's book (1952) for an enter- 
taining pictorial account of the appearance of symmetry in 
Nature. 



T 



tensor. To a mathematician a tensor is an object that trans- 
forms in a particular way when the coordinates he is using to 
describe a problem are transformed. A scalar quantity, which 
may be regarded as a zeroth-rank tensor, does not change 
when the coordinates are transformed. A vector does change 
when the coordinates change (at least, the 'object' remains 
the same, but its description in terms of the new axes is 
different from its description in terms of the original axes) : 
it is a first-rank tensor. More complex objects may be given 
as examples of higher-rank tensors; for example, the object 
rr, where r is a vector, is a second-rank tensor. The mathema- 
tician distinguishes between contravariant and covariant 
tensors: for example, the set of coordinates [x, y, z] of a 
vector transforms as a contravariant tensor of rank one, and 
the set of objects [df/ax, bfloy, bflbz] , where r"is a scalar 
function, transforms as a covariant tensor of rank one. Tensor 
calculus in its most powerful form is concerned with general 
transformations of coordinates, and is a powerful tool when 
non-euclidean geometry has to be studied: for instance, in 
general relativity. When the transformations are limited to 
orthogonal transformations, those preserving angles in the 
transformation (like a simple rotation of axes), the proper- 
ties that an object must possess to be a tensor are less 
stringent, and the objects that comply are termed cartesian 
tensors. When the objects transform in a special way under 
rotations we encounter the irreducible spherical tensors; 
these are of considerable importance in the discussion of 
molecular and atomic properties. 

To a chemist a tensor is a quantity that expresses the 
directional dependence of the properties of molecules and 



solids; he is normally on safe ground because usually, but not 
always, the properties that interest him transform in the 
same way as tensors, and so are tensors. As an example 
consider the effect of an electric field on a molecule. Let the 
field be along some axis Z, then the field "polarizes the 
molecule and induces a dipole moment. Usually the major 
component of this induced dipole lies in the same direction 
as the field, but in general the field and the induced dipole 
are not exactly colli near, and there are also X- and V- 
components (Fig. T1 ). The magnitudes of the components 
are proportional to the strength of the applied electric 
field E z , and so the dipole has components a E , 

a yz^z' anc ' a zz^z' wnere tne a QQ- are tne coefficients of 
proportionality, the "polarizability. If the field is applied 




FIG. T1. The effect of off-diagonal elements of the polarizability 
tensor. When E is along an arbitrary direction it may induce a dipole 
which need not be parallel to E. The dipole is parallel if E lies along a 
principal axis (one of the axes marked on the molecule). 



235 



236 



terms 



along the X-or V-axes similar expressions could be set up 
to describe the induced dipole (which may have different 
components if the molecule is anisotropic}. In all we g et / 
nine quantities of the form « QQ ,. Recall that the quantity 
rr, with the nine components XX, XY, XZ, . . . ZZ is a 
second-rank tensor; then we see that by analogy we may call 
the nine quantities a QQ , the components of a second-rank 
tensor a, the polarizability tensor. Knowing a we may predict 
the polarization in any direction when a field is applied in 
any direction. 

Other physical quantities that behave like tensors include 
those representing a strain a stress type of relation, and it is 
from this that the name tensor arose. In the example just 
described the stress is supplied by the applied field, and the 
strain is the resulting polarization. The -magnetic suscep- 
tibility is of the same type, and may be expressed as a second- 
rank tensor. The elasticity of a body is of this form, and it 
too may be expressed tensoriaily. For rather special reasons 
the j-tensor of "electron spin resonance is not a true tensor, 
and therefore we have called it the ^-value; this is a slight 
pedantry because most people do call it a tensor, which in 
many cases it very nearly is. 

Aspinor is a quantity that resembles a tensor, but has the 
peculiar property that when it is rotated through 360° it 
changes sign; therefore a rotation of 4rr must be made to 
bring it back to its initial form. Objects such as this are in- 
volved in the mathematical description of spin; hence the 
name. 

Further information . An excellent and moderately simple 
introduction to the general mathematical theory of tensors 
will be found in Synge and Sen i Id (1949); see also Kyrala 
(1967) and Jeffreys (1931). The role that tensors play in the 
discussion of the physical properties of matter is described 
by Brink and Satchler (1968), Rose (1957), and Edmonds 
(1957). See Fano and Racah (1959) for a consolidated 
account. Why g is not a tensor is described in §15.6 of 
Abragam and Bleaney (1970). 

terms. The word 'term' first entered spectroscopy with its 
colloquial meaning, and therefore without much fundamental 
significance. In the early days of atomic spectroscopy it was 



found that a" considerable simplification of the description of 
the spectrum of an element could be obtained by expressing 
the frequency of every line as the difference of two terms: 
since one term contributed to a number of spectral lines it was 
necessary to list far fewer term energies than transition 
energies. The rule that the spectrum could be so expressed is 
known as the Ritz combination principle (terms are combined, 
by subtraction, to give the frequency or energy of a transition). 
Thus initially the 'terms' of a spectrum were just a collection 
of numbers which could be used to predict spectra; but, rather 
oddly, it was found that not all combinations (7"j - 7~ 2 ) 
corresponded to an observable transition. 

The quantum theory provided a natural interpretation of 
terms: it revealed that a term was an energy state of the atom, 
and the combination principle was elucidated partly by "Bohr's 
postulate and later by proper quantum-mechanical calculation 
which showed that a spectral line represented the transition of 
the atom from one stationary energy state to another, and that 
the frequency was determined by hv=T\ — T<i. Quantum 
mechanics also permitted the calculation of "selection rules, 
which tell whether a given combination of terms is observable. 

A term is now interpreted as an energy state of an atom, and 
a term symbol, which labels the energy state, may be con- 
structed from the quantum numbers that define the energy 
state. A typical term symbol looks like 2 S or 3 P (but other 
indices are often attached for a finer description of the state, 
as we shall describe). The symbol, illustrated in its full glory 
in Box 26, is constructed as follows. 



BOX 26: Term symbol 



multiplicity 
value of I — 



(2S+1) M 



— state 
level 



L = 0, 1 , 2, 3, 4, . 
X = S, P, D, F,G, 



1. The letter denotes the value of L, the total "orbital angular 
momentum quantum number, according to the correspondence 
L = 0, 1, % 3,4, . . .+-+ S, P, D, F, G 

2. The left superscript is the value of 2S + 1 , where this S is 



torsional barriers 



237 



the total 'spin angular momentum quantum number for the 
atom {a massive intellect is not required in order to avoid 
confusion of the S with the S of L = 0), The value of 2S + 1 
is often referred to as the 'multiplicity of the term, and the 
propriety and significance of this are described under that 
entry. Thus 2 S is read as a 'doublet S term', and 3 P as a 
'triplet P term'. 

3. The 'levels of a particular term are distinguished by a right 
subscript of the value of J; for example, a doublet term with 

L = 1 (that is, 2 P) has two levels with J= jor §: the levels are 
denoted 2 P l/2 or 2 P Vi , respectively, and read 'doublet P 
one-half or 'doublet P three-halves'. 

4. The state of an atom can be expressed in even greater 
detail if one also quotes the value of M Jt the projection of J 
on some axis: this label is added as a right superscript. For 
example the Mj = — 1 state of the J = 2 level of a triplet 
term with L = 2 would be labelled 3 D 2 ~' , and from this 
symbol we could write the values of the quantum numbers 

5. L,J, and/W,[seeBox26). 

A list of terms and their energies enables the spectrum to 
be reconstructed; but to do so it is also necessary to know 
the permitted combinations: these are determined by the 
"selection rules which are normally expressed as the values 
of AS, AL , AJ, and AMj permitted in a transition. Inspection 
of the term symbol in conjunction with these rules enables 
one to see very clearly which transitions are allowed. 

Term symbols are also encountered in molecular spectroscopy, 
but as L is then not a good quantum number the labels usually 
denote some "group-theoretical classification. 

Further information. See MQM Chapter 8 for a discussion of 
atomic spectra and Chapter 10 for molecular spectra: both 
Chapters illustrate the use of term symbols. Further infor- 
mation will be found under "atomic spectra, "selection rules, 
"fine structure, and "multiplicity. 

torsional barriers. The classic example of a barrier to free 
rotation is that in ethane: one methyl group cannot rotate freely 
about the CH; bond because of its interaction with the other 
methyl group. The lowest-energy conformation is the 
staggered arrangement of the two methyls, where a view along 
the C— C bond shows that the C— H bonds of one group bisect 



harmonic 
Ceo sine) 

potential 




FIG. T2. The torsional barrier in ethane, and the fit of a harmonic 
potential for small librations. 



deep 

oscillator 
ocm 




FIG. T3. Energy levels of free rotor (on the )eft> are modified by a 
periodic potential and in the limit of a deep potential become those of 
two independent oscillators. Note how the degeneracies change, q - 
, wtwe V is the depth of the well. 



iV/211 1 , 



238 



torsional barriers 



the H— C— H angles of the other; the highest-energy confor- 
mation is the eclipsed, when the C— H bonds of each group 
are in opposition. If one methyl group is twisted through a 
smalt angle from the staggered conformation, and then 
released, it will execute torsional oscillations. As more energy 
is added to this twisting mode the oscillations get larger, and 
soon the methyl group is able to jerk from one energy 
minimum to another. When sufficient energy has been added 
the group rotates in a manner that is almost indistinguishable 
from a free rotation. Thus a torsional vibration becomes a 
free rotation. Conversely, a free rotation becomes localized as 
a torsional oscillation as an energetic molecule loses energy. 

The quantum-mechanical explanation of the process treats 
the system as a particle in a potential which varies as cos(30/2) 
{see Fig, T2): this has the correct periodicity of the actual 
system but is only a moderately good approximation to the 
actual dependence on the torsional angle 8 (which in detail 
is very complicated). At the troughs of the cosine function the 
potential vaies 3S 6 2 (Fig, T2), and so a particle confined close 
to the minima behaves like a "harmonic oscillator. It follows 
that the ground state of the ethane torsion is a zero-point 
rocking about the bond. The first excited state of the torsional 



mode resembles a harmonic oscillator in each of the wells; but 
the correspondence is exact only for infinitely deep potentials. 
In the real molecule the methyl group can "tunnel from one 
well to another, and so the wavef unctions are not truly those 
of a harmonic oscillator. 

At high excitations, when the total energy greatly exceeds 
the peaks of the barrier potential, the potential can be ignored. 
In this limit the group behaves like a particle of mass 3m 
confined to a ring. 

The mathematics of the situation can be illustrated nicely 
if we turn from the methyl group to an example where there 
are only two wells, not three (Fig. T3). For deep wells every 
energy level is doubly "degenerate because the rocking group 
is in either of the two equivalent wells; for vanishingly shallow 
wells every level except the lowest is doubly degenerate because 
a free rotor may rotate clockwise or anticlockwise with the 
same energy. (The lowest level, corresponding to a static rotor, 
is non-degenerate. ) As the barrier is strengthened the latter 
situation must pass over into the former. This may be demon- 
strated explicitly, because the Sch rod inger equation with a 
harmonic potential is a Mathieu equation, and its solutions 
are tabulated. In Fig, T3 we show the energies and in Fig. T4 




FIG. T4. Torsional wavefunctions for shallow and daep walls: note how the functions are more those of a rotor in the former, and. independent 
oscillators in the latter. 



transition probability 



239 



the wavefunctions for barriers of different heights. These 
pictures illustrate the preceding discussion, for the emergence 
of the free-rotor energies and wavefunctions is clearly apparent. 

Further information , Helpful discussions of the hindered- 
rotation problem are given in §8.3 of Sugden and Kenney 
(1965) and §12.6 of Townes and Schawlow U 955). Solutions 
and properties of the Mathieu equation are given in §20 of 
Abramowitz and Stegun (1965), and they give numerical data. 
See also "inversion doubling for a related situation. A review 
of the internal-rotation problem has been given by Strauss 
(1968). 

transition probability. A transition probability is the prob- 
ability that a system will change from one state to another. Its 
principal significance is that it determines the intensity of 
spectral lines and so, for poets, the brilliance of a colour. The 
quantum -mechanical discussion of transition probabilities is 
normally based on -perturbation theory, for the application to 
the molecule of an oscillating electromagnetic field distorts it 
slightly, and this distortion can be interpreted as an admixture 
of excited states into the original state. Since the *wavefunction 
now contains a component of the excited states there is a 
non-vanishing transition probability. 

The transition probability depends on the strength of the 
light (the energy of the perturbation), on the match between 
the frequencies of the exciting radiation and the excited 
transition (for the coupling is strongest when there is 
'resonance), and on the strength of coupling between the 
molecules and the electromagnetic field. The last is related to 
the transition dipole (see "electric dipole transition), which is 
essentially a measure of the extent of movement of the charge 
during the transition: if the motion is great, then the field can 
interact strongly, and the spectral line is intense. 

As an example, consider the excitation of that well-worn 
species the hydrogen atom. The 2p ; *— 1s excitation occurs 
with a movement of charge from a spherical distribution 
about the atom to one clustered around the z-axis; the 
transition can be pictured as an oscillation of charge backwards 
and forwards along the z-axis, and the dipote moment associ- 
ated with this motion ensures that the line appears brightly in 
the spectrum. The 2s «— Is transition, on the other hand, 



although it involves a significant migration of charge, does so 
with a conservation of its initial spherical symmetry, and so 
there is no dipole moment associated with the oscillation from 
one state to the other; the light field does not interact, and the 
transition does not occur in the spectrum. 



BOX 27: Transition probability 

Fermi's Golden Rule 

Transition rate (rate of change of probability) for a 
migration from a state / to a state f under the influence 
of a perturbation of strength V is 

where V fj = {\p f \V\\jj.} and p{v f .) is the energy density of 
incident radiation at the transition frequency V... 
Electric dipole transitions 

B fr \d f A 2 /6e tf = B ff . 

8 is 'Einstein's coefficient of stimulated absorption or 
emission. The total emission rate is 

A ff =Bi!h(v fj lc} 3 B fj in general 

= (8TrV3eoC 3 h)i£.ld ft .P for electric dipoles. 
Transition dipole 

d f . = fdr\j/*d \p. d = -er. 



Fermi's Golden Rule is the mathematical expression for the 
determination of a transition probability. It is set out in Box 27, 
and the form quoted may be interpreted as the product of the 
square of the transition dipole moment between the states (the 
dipole acts as a handle for the modification of the amplitude 
of the wavefunction, and to get the probability the square must 
be taken) multiplied by the energy density of radiation at the 
transition frequency (the more powerful the stimulus the greater 
the transition probability). The formula may be deduced from 
time-dependent -perturbation theory. 



240 



tunnelling 



This account of transition probabilities underlies the dis- 
cussion of "selection rules, which enable one to predict when 
a transition probability is non-zero. The strength of the 
transition dipole is often quoted in terms of the "oscillator 
strength, or in terms of the -Einstein A and 8 coefficients. 
These are conveniently related to the "extinction coefficient, 
which is an experimental measure of the spectral intensity. 

Questions. 1 . What does the transition probability determine? 
Why does the application of a time-dependent perturbation 
induce spectral transitions? At what frequency is the absorption 
strongest? What is a transition dipole moment? Is there a non- 
vanishing transition dipole moment associated with the 
following transitions: 2p y « — 1s, 3p v * — 1s, 3d„„ « — 2p„ 

2p. 



r x "' x ' xy x 

2p , 3p * — 2p ? Consider the transition moments 

associated with the emissions 2p x — * 1s and 2p^ — > Is: would 
you expect the polarization of the emitted light to be different 
in the two cases? 

2. The x-component of the transition dipole moment d* f . for 
the transition from a state described by a function \j/ / to one 
described by a function \jj f is equal to the integral ~6JdT\pfX\jJ., 
with similar expressions for the y- and z-components. The 
transition probability is proportional to Id^i*. Show that the 
stimulated absorption intensity for a particular transition is 
equal to trie stimulated emission intensity. Calculate the trans- 
ition dipole for the transition from the ground state of a particle 
in a one-dimensional square well to the first excited state 
(use the information in Box 15 on p. 166). From the infor- 
mation in Tables 11, 14, and 15 calculate the intensity of trans- 
itions from the ground state of a one-dimensional harmonic 
oscillator to the first and second excited states, and for the 
transition 2p < — 1s of the hydrogen atom. 

Further information. See MQM Chapters 7 and 11. Transition 
probabilities are described by Eyring, Walter, and Kimball 
(1944), Heitler [1954), Kauzmann (1957), and the standard 
works on quantum theory; see Davydov (1965), Schiff (1968), 
and Messiah (1961). Transition probabilities occur in the 
discussion of "selection rules, "oscillator strengths, "Einstein 
A and 8 coefficients, "polarizabilities, and "intermolecular 
forces. A nice, simple discussion will be found in Loudon 
(1973). 



tunnelling. Quantum-mechanical tunnelling is the process 
whereby particles penetrate potential barriers and appear in 
regions forbidden to them in classical mechanics. Consider a 
■particle trapped in a square well by a potential barrier of 
finite height and width. Let the particle have an energy which, 
according to classical mechanics, is insufficient to carry it over 
the potential barrier. The quantum mechanics of the situation 
shows that the particle has some probability of tunnelling 
Through the barrier and escaping from the well. The tunnelling 
probability arises from the requirement of the continuity of 
the wavef unction at the walls of the well: if the wavefunction 
has a non-zero amplitude at the inside edge of the barrier 
{and this is permitted when the potential there does not 
become infinite) it cannot simply vanish within the barrier; 
instead it will start to decay more or less rapidly towards zero 
(Fig. T5). If the decay of the function is not too rapid the 



porertrinl barrier 




FIG. TS. Tunnelling of a particle from left to right through a barrier. 

amplitude might not have reached zero at the outer edge of 
the barrier. At this point the function must butt smoothly on 
to the function characteristic of the free particle beyond the 
barrier, and from then on the wave is essentially undiminished. 
Since the wavefunction of the particle does not vanish in the 
region outside the barrier there is a non-vanishing probability 
that the particle will be found in this region, and so it has 
tunnelled out of the original well. 

The wavefunction inside the barrier itself falls more quickly 
as the height of the barrier is raised (relative to the energy of 
the incident particle), and so the amplitude of the function on 
the outside of the barrier decays correspondingly quickly. This 
implies that the tunnelling probability diminishes rapidly as the 
barrier height is increased. The amplitude also decays more 
rapidly as the mass of the particle is increased: therefore the 
tunnelling diminishes as the mass of the tunneller increases. The 



tunnelling 



241 



shape of the potential is also important because sharply changing 
potentials reflect the particle more effectively than slowly 
varying potentials (an analogous situation occurs in the propa- 
gation of light: light is reflected most strongly from regions 
where the refractive index changes abruptly). Therefore the 
most favourable tunnelling situation is that of a light particle 
confronted with a slowly varying potential barrier. Electrons 
tunnel very effectively; protons tunnel much less well, but still 
appreciably; deuterons tunnel little. 

Some concern is often expressed about the apparently 
nonsensical occurrence of a negative kinetic energy in the classi- 
cally forbidden regions, where the potential energy apparently 
exceeds the total energy. The occurrence of a negative kinetic 
energy is indeed the reason for the exclusion of particles from 
these regions in classical mechanics, but it is no problem in 
quantum mechanics because the "kinetic energy of a particle 
must be interpreted as the mean value evaluated over the 
entire wavefunction. The fact that the kinetic energy is locally 
negative does not interfere with the fact that the measurable 
kinetic energy, the mean value, is positive. Furthermore, 
suppose we were to attempt to confine the particle into the 
region of the barrier in order to force it to have negative kinetic 
energy: with drums sounding and bugles blaring the "uncertainty 
principle gallops to its aid; for the very act of confining the 
particle to a particular region introduces an uncertainty into 
the energy, and this uncertainty is sufficient to prevent us from 
concluding that the particle has a negative kinetic energy. 



Questions, 1 . What is meant by tunnelling? What are the most 
favourable conditions for it? What is its quantum-mechanical 
explanation? Why is it unnecessary to worry about negative 
kinetic energies? Discuss the tunnelling probability for particles 
that are fired with gradually increasing kinetic energy against a 
rectangular barrier. What happens when the kinetic energy is 
enough, according to classical mechanics, to take them cleanly 
over the top of the barrier? 

2. Consider a rectangular barrier of height V and width W; on 
either side the potential is zero for all space. Set up and solve 
the "Schrodinger equation for this system, and from the 
continuity of the wavefunctions at the edges of the barrier 
construct acceptable solutions. Show that the probability of 
tunnelling depends strongly on the values of V and W, and on 
the mass of the particle. Calculate the relative probabilities of 
an electron, a proton, and a deuteron tunnelling through the 
barrier. 

Further information . Calculations on tunnelling phenomena 
are described in Chapter 1 1 of Bohm (1951), §26 of Davydov 
(1965), Messiah (1961), Schiff (1968), and other standard 
works. Problems and worked solutions will be found in 
Gol'dman and Kryvschenkov (1961). Important applications 
of tunnelling are in the "photoelectric effect, and so see that 
entry, in chemical reactions— see Harmony (1972) for a 
review— and in electrode processes; see Albery's Electrode 
kinetics (OCS 14). 



u 



uncertainty principle. The uncertainty principle reveals that 
there exist pairs of observables to which it is not possible to 
ascribe simultaneously arbitrarily precise values: as one observ- 
able is constrained to have a more precise value so its conjugate 
partner becomes more ill-defined. An experiment set up to 
determine the two observables simultaneously is able to 
determine one precisely only at the expense of losing infor- 
mation about the other (see -duality); and the product of the 
uncertainties in the two simultaneous measurements can never 
be less than a small but non-zero value of the order of h. 

The most famous example of this situation is the uncertainty 
inherent in the simultaneous determination of the position of a 
particle along some coordinate q and its component of linear 
momentum along the same coordinate p. These two observ- 
ables are conjugate in the sense of the uncertainty principle, 
and if we can pin down the position to within a range hq 
(where 8q is actually the root-mean-square spread of the 
particle's location about some point) then the uncertainty 
principle demands that the value of p must be uncertain to 
the extent op (again this is a root-mean-square spread) such 
that the product of uncertainties 5q6p does not have a value 
of less than jm As the position is ascertained more sharply 
(and 8q decreased) the spread of p must increase in order 
to ensure that the uncertainty product &q$p does not fall 
below s h. Conversely, if we were prepared to forgo all infor- 
mation about the particle's momentum so that op could be 
allowed to become indefinitely large, then Sq could be made 
indefinitely small, and the position determined with arbitrary 
precision. Unfortunately the implication is that, although we 
now know the position of the particle with arbitrary precision, 



it is not possible to predict where the particle will be at any 
instant later, as we can know nothing of the particle's 
momentum at the moment of determining the position. Thus 
the uncertainty principle eliminates the concept of a trajectory, 
a concept central to classical mechanics. 

Alternatively we could measure the momentum p with 
arbitrary precision, but in order to preserve the uncertainty 
product we should be forced to forgo all information about 
the position q: this approach also eliminates the concept of 
trajectory. 

It should be noticed that the uncertainty principle for 
position and momentum refers to components along the same 
axis and puts no restriction on the simultaneous values of these 
observables along different axes. Consequently the position 
along x may be measured simultaneously with the momentum 
p along y, and there is no inherent limitation on the precision 
of the determinations. The mathematical expression of the 
uncertainty principle (see Box 28) enables us to decide which 
observables are conjugate; but a rough guide is that conjugate 
variables consist of the coordinate and the momentum 
corresponding to that coordinate. 

Discussions of the uncertainty principle are often put in the 
form of presenting a duffer (and long live all such duffers) who 
attempts to do an experiment which will deny the predictions 
of the principle; he retires, of course, bruised from the ring. 
Heisenberg, whose principle this is, presented such a duffer In 
order to show that all such gedanken experiments (thought 
experiments) must fail. His jester used a microscope to measure 
the position of the particle and, in order to do so with increasing 
precision, selected one operating with ever shortening wave- 



242 



u ncerta i nty pr i n ci pie 



243 



BOX 28: Uncertainty principle 

Let A and B be the "operators corresponding to the 
observables A and B, and let 64 and 5fi be the r.m.s. 
deviations from the mean: 

54=[C4 a )-W> 2 ) M , &B=l<B*)-<B) 2 ] K . 

According to the uncertainty principle, these must satisfy 

Sa5b>Ikia,b))\, 

where [A. B\ = AB - BA, the -commutator. 
Typical uncertainty products include the following: 

See text for significance of T. 



length of light. But the shorter the wavelength the more 
momentum each photon carries (-de Broglie relation), and 
since at least one photon must be scattered into the micro- 
scope aperture in order for the position to be determined, it is 
clear that the very act of observation imparts a momentum to 
the particle. An analysis of the experiment, taking into account 
aperture-diffraction effects and momentum transfers on light 
scattering, concludes that the uncertainty product &jr6p is 
indeed not less than jh (h enters through the de Broglie 
relation). In a classical world the jester would laugh last, 
because h would be zero and there would be no intrinsic limi- 
tation on the precision. 

Thought experiments of this nature illustrate at an obser- 
vational level what the uncertainty principle reveals about the 
nature of matter at a much deeper level. Momentum and 
position are linked by the interpretation of the "wavef unction. 
A system in a state of well-defined linear "momentum is 
described by a plane wave of well-defined wavelength; but this 
wave, which for a momentum kb can be written exp \kx, 
corresponds to a probability distribution proportional to 
I exp \kx | 2 , and this is independent of x. Therefore a state 
of well-defined momentum describes a particle with a com- 
pletely undefined position. Conversely, in order to describe 
a localized particle, a "wave packet must be formed with an 
amplitude large at one point and small elsewhere. This can be 



achieved by "superimposing a large number of waves of 
different lengths, and therefore of different momenta. Conse- 
quently, the sharper the wave packet we try to form (in order 
to get a more localized particle) the wider the range of 
momenta of the particle. 

The other pairs of conjugate observables can be found by 
testing whether the "commutator of their corresponding 
"operators disappears: if it does not, the observables cannot be 
determined simultaneously with arbitrary precision; if it 
vanishes there is no restriction. Some important pairs of 
conjugate observables are listed in Box 28. 

The energy -time uncertainty relation differs from the rest 
in a subtle way: there is no operator for time in quantum 
mechanics (it is a parameter, not an observable), and so the 
commutation rule cannot be applied. The relation should be 
viewed as a consequence of the lack of commutation of the 
position and momentum operators, or equivalently, as a 
consequence of the Schrridinger equation. The energy -time 
relation depends upon the existence of an evolution of the 
system with a characteristic time t; when such a process is 
present the energy levels of the system are indeterminate by 
an amount 6f such that the product t8E does not fall below 
^h. For stationary states, where r is infinite, the energy may 
be defined with arbitrary precision; but where a state has a 
finite lifetime its energy is correspondingly imprecise. 

A final word may be said on cyclic systems: the uncertainty 
relations for angle and angular momentum must be treated 
with care because an uncertainty of 2ir in angle is equivalent 
to complete uncertainty: special forms of the uncertainty 
principle are used in these cases. 

Questions. 1. State the uncertainty principle. Discuss the 
principle as applied to the determination of the position x and 
the momenta p x and p , May the position coordinates (jk, y, z) 
of a particle be specified simultaneously with arbitrary 
precision? May the kinetic energy and the momentum of a 
particle be specified simultaneously? Why does an experiment 
to determine the position of a particle interfere with the 
momentum of a particle? How does the wave nature of matter 
illuminate the connexion between the position and the 
momentum of a particle? Why does the wave picture allow 
x andp to be determined with arbitrary precision? Why is 



244 



united atom 



the concept of trajectory alien to quantum mechanics? Why is 
the energy-time uncertainty relation peculiar? 

2. The position q of a particle is determined to within a range 
0-1 mm, 1 ilTt, 1 nm, 1 pm; what is the corresponding simul- 
taneous uncertainty in the momentum p? If the particle is 
an electron, to what kinetic energy does the uncertainty in 
momentum correspond? Three states in an atom decay with 
time constants Ots , 1 jus, 1<T t2 s; what is the uncertainty in 
the energy of the atom in each excited state? The natural 
width of spectral lines is determined by the lifetime of states, 
as suggested by the last part of this question. See "electron 
spin resonance. 

3. Use the mathematical expression of the uncertainty 
principle as set out in Box 28, to investigate the limitation on 
the simultaneous determination of the following pairs of 
observables: x aitdp ;x and p ;p x andp ; fi^ and x ; z and S! ; 
kinetic energy and Coulomb potential energy; total energy and 
x; total energy and dipole moment. 

Further information. See MQM Chapter 4 for a deduction and 
application of the principle. For a discussion of the uncertainty 
principle see Heisenberg (1930) and Ingram's Radiation and 
quantum physics (OPS 3), and the standard quantum-mechanics 
texts such as Dirac (1958), Messiah (1961), Schiff (1968), 
Davydov (1965), and Landau and Lifshitz (1958a). For an 
interesting account laced with speculation on mind and magic 
see Bohm (1951). For an account of one aspect of the energy- 
time relation see Salem and Wigner (1972), and for an account 
of the uncertainty principle for cyclic systems see Carruthers 
and Nieto (1968). 

united atom. The united-atom method, which is used to 
describe the structure of molecules, is one of a variety that 
employs a correlation diagram. In general, one has a set of 
orbitals or states of a system when it is in one form and a 
set of orbitals or states for the system when it is in another 
form, and one is interested in which states of one form turn 
into which states of the other, A correlation diagram consists 
of two arrays of levels joined by lines which denote the way 
that a state changes into another state when the system 
changes from one form to another. The most important rule 
for constructing such a diagram is that lines representing 



states with the same symmetry cannot cross (see "non-crossing 
rule). We illustrate the technique with the idea of the united 
atom; this is a lineal ancestor of the Walsh diagrams, which 
show how the molecular orbitals of molecules change when 
bonds are bent, and both are parents of the Woodward-Hoffman 
rules, which show how molecular orbitals and states change 
during concerted molecular rearrangements. 

The united-atom correlation diagram takes as one set of 
states the "orbitals of two separated atoms (we let the atoms 
be the same and call them A), and considers how these orbitals 
change as the atoms are pressed together until ultimately they 
fuse into an atom of twice the atomic number, the united 
atom. The energy levels of the united atom are known, so are 
those of the separated atoms; what is unknown is the structure 
of the intermediate object, the diatomic molecule A-A. 

Let us take as the simplest illustration of the method the 
atoms of hydrogen; the united atom will therefore be the 
helium atom. Whether or not the fusion can actually be made 
to occur in practice is important for the future of mankind 
but immaterial for the present discussion. Concentrate on the 
1s-orbitals of the separated atoms, and envisage what happens 
as we squeeze the atoms together. If the atomic orbitals are 
squeezed together in phase (so that a positive-going amplitude 
*of one overlaps a positive-going amplitude of the other) the 





united atom 



molecule 



separate atoms 



FfG. U1. The united atom (He) formed from H + H, and the inter- 
mediate Hj. 



united atom 



245 



process of uniting the atoms ultimately generates the Is-orbital 
of the helium atom, (See Fig. U1 .) The out -of -phase super- 
position of the two atoms always possesses a 'node half-way 
between the two merging nuclei: when the nuclei are united, 
half-way between them means through the united nucleus; 
therefore the merging of the orbitals has generated a 
2p-orbital of helium (Fig. U1). This is known to lie above the 
1s-orbital, and therefore we may conclude that at an inter- 
mediate stage the hydrogen molecule possesses the orbitals 
denoted a and 0*. These are just the bonding and an ti bonding 
orbitals of 'molecular -orbital theory, and to them we may 
apply the -aufhau process in the normal way, and so arrive at 
the structure of the molecule. 

This process may be applied to a more complicated pair of 
atoms in order to arrive at some assessment of the structure of 
the intermediate molecule. There are some difficulties, and 
these are mainly connected with the role of spin -orbit coupling 
in heavy atoms, for this serves to muddle some of the 
correlations. 

Questions. What is meant by a "correlation diagram'? In what 
sense is the united-atom procedure an example of the use of a 
correlation diagram? To what use may the united-atom 
procedure be put? In what ways does the correlation diagram 
drawn in the figure differ from the actual dependence of the 
energy of the orbitals on the internuclear separation? Do 
these differences matter? What rule has to be observed in the 



formation of a diagram? Construct the full correlation diagram 
for the formation of a united atom from two atoms possessing 
1s-, 2s-, 2p-, 3s-, and 3p-orbitals, and use it to discuss the 
electronic structure of the homonuclear diatomics that may be 
formed from the first-row atoms. (Check your answer against 
Fig, M8, which shows the intermediate situation corresponding 
to the structure of 2 .) 

Further information. A nice discussion will be found in §4.7 of 
Coulson (1961), and an original paper on the subject is that of 
Mul liken (1932). See also Chapter VI of Herzberg (1950) for a 
thorough discussion of the way the united atom is used to 
discuss the structure and spectra of diatomic molecules. A rule 
of special importance is the Wigner-Witmer spin-correlation 
rule, which tells how to determine which atomic states are 
formed when a diatomic breaks up. This is discussed in 
Herzberg (loc. cit.), §5.2 of Wayne (1970), and Chapter 3 of 
Gaydon (1968). The Walsh rules for the discussion of molecular 
structure are given in a classic series of papers by Walsh (1953). 
What these diagrams are diagrams of has been the cause of 
much perplexity: for a readable analysis consult Coulson and 
Deb (1971 ). The Woodward-Hoffman rules are described by 
Gill (1970), Woodward and Hoffman (1970), Gill and Willis 
(1969), and Longuet-Htggins and Abrahamson (1965). See 
also Woodward and Hoffman (1969, 1970) for a review with 
many applications and also Alder, Baker, and Brown (1971) 
for a helpful description. 



V 



valence bond. The valence-bond theory was the first 
quantum-mechanical theory of the chemical "bond, and 
drew heavily on the chemist's concept of a bond as an 
object depending for its strength on the presence of two 
paired electrons. The theory picks out of a molecule the 
electrons that are paired (the perfect-pairing approximation) 
and supposes that these dominate in the formation of the 
bond; when several perfect-paired structures have similar 
energies the molecule is allowed to "resonate among them, 
and the energy of the whole is thereby lowered. 

As in most things, the simplest object can elucidate the 
method most effectively. Consider H 2/ that most public of 
molecules. At great separations the "wavefunction for the 
species is the wavef unction for the two separated atoms 
V> (1) ^ b (2), which for brevity we shall denote a\b 2 - When 
the atoms are as close as they are in the molecule the wave- 
function might not differ very greatly, the only difference 
being that we cannot stop one electron slipping off its 
nucleus and visiting the other. In other words we must permit 
the electrons to exchange their roles. In terms of the wave- 
function we must \eta l b 2 be contaminated by a 2 b x , in which 
electron 2 occupies the orbital hitherto occupied by electron 
1 , and vice versa. In fact, from the symmetry of the system, 
the contamination must be allowed to proceed so far that the 
wavef unction is a 50:50 mixture of both arrangements 
{a\b-i +a 2 bi). At this point we sit back, the physics having 
been done, and do the mathematics. This means that we 
attempt to calculate the energy of the molecule from the 
wavefunction we have set up. 



There are some tricky integrals over the coordinates of the 
electrons that need to be done in order to evaluate the 
potential energy of the molecule, and they are illustrated in 
Fig. VI , One integral can be interpreted as the contribution 




j'=/d^, a a)V/4Tr£g 







FIG. VI. Contributions to the v.b. energy of H 2 . 



246 



valence bond 



247 



enerWau 




FIG. V2. The dependence on (he nuclear separation of the mo I ecu I ar 

integrals contributing to the VB energy of H^ The total energy is also 
shown. 

to the total energy from the attractive interaction / between 
an electron on one nucleus and the other nucleus; another is 
the repulsive interaction / between the electron clouds on the 
two nuclei. An analysis of the electron distribution for the 
wavef unction a^ +a 2 6, shows that there is a significant 
accumulation of electron density in the internuclear region- 
this extra accumulation of density is represented by the oval 
shapes in Fig. VI— and this extra density contributes extra 
terms to the total energy. One contribution is the repulsive 
interaction k between the two electrons confined to the oval 



BOX 29: Molecular integrals and experimental data for H 2 

S = /dT (3(1)60) 

/ = SdTidr 3 «(1 ) 2 (e 7 /4morn)b{2\ 2 
/^fdT ia (\) 2 (e 2 /4m^ ib ) =/dri6(2) 2 (e 2 /47re r 2a ) 
k = fdr, dr 2 a( 1 )b{ 1 )ie 2 /ATre r 11 )a<2>o(2) 
*' = /dTi3(1)6(1 )(* a /4nEoT ,„) = fdT 2 a(2)b(2) \fi 2 fA,iK^ u ). 

Coulomb integral J = {e 2 f4m R) +/" — 2/'. 

Exchange integral K = ie 2 S 2 /4ire R) + k- 2Sk'. 

Energy E ± = 2E U (H) + { jff.} (£, < SJ. 

Fi,(H) is the energy of an isolated hydrogen atom. 
Bond length inH 9 : 74- 16 pm (0-7416 A). 
Dissociation energy (£>g): 4-476 eV, 36 116 cm" 1 . 
Rotational constant: 60809 cm" 1 . 
Vibrational frequency: 4395-2 cm" 1 . 



regions, and another is the attractive energy k' between these 
electron -rich regions and the nuclei {see Box 29 and Fig. V2). 
Just looking at the numerical values of the integrals shows that 
the most important contribution is the last: in this model the 
reduction in energy of the molecule below the energy of the 
separated atoms is in large measure due to the lowering of the 
potential energy of the electrons by virtue of their accumulation 
in the internuclear region, where they are able to interact attract- 
ively with both nuclei. (We note and emphasize that an accurate 
account of the source of bonding energy must also take into 
account the changes in -kinetic energy of the electrons on bond 
formation and the distortion of the atomic orbitals in the 
vicinity of the nuclei: see °bond and "molecular orbitals.) 

The numerical value of the bonding energy for the hydrogen 
molecule calculated in the manner described is 3-14 eV, in 
moderate agreement with the experimental value 4-72 eV. (In 
fact, the agreement is bad, but when it was first obtained the 
number supported the view that the model was correct; it has 
since been made much better without altering the essential 
features of the description.) 



248 



valence bond 



This approach to hydrogen is developed to account for the 
structure of more complicated molecules, but the procedures 
rapidly become more complex. In each case the bonding is 
ascribed to the interaction of pairs of electrons, and so pairs 
of electrons are selected and the energy of their interaction is 
calculated. One of the important features of the valence-bond 
theory now appears: because there are many electrons in an 
atom, and therefore many pairs, we have to take into account 
the possibility that electrons that form different paired bonds 
still interact electrostatically. In this connexion we may think 
of "benzene and one of its Kekule structures. In the structure 
the spins are paired when TT-bonds are formed, and so the con- 
tribution of each bond to the energy of the structure arises in the 
same way as we discussed for the hydrogen molecule. But the 
electrons in one bond interact with the electrons in the others. 
This gives rise to two effects. First, the energy of the Kekule 
structure Is modified; and second, there is a tendency for the 
electrons to redistribute themselves around the ring (see 
Fig. B8 on p. 19 or Fig. R7 on p. 201 ). One redistribution 
corresponds to the other Kekule structure, and so the effect 
of the interaction is to induce "resonance between the two 
Kekule structures. This alters the energy. The true distribution 
of electrons in the ring cannot be described simply by the 
resonance of two Kekule structures: a better description and 
a lower energy are obtained if other canonical structures are 
allowed to take part in the resonance (see 'benzene). 

The principles described in the preceding paragraphs are 
the bases of the general vafence-bond method. One selects the 
basic perfect-paired structures according to a set of rules (and 
so they are called canonical; canon = rule), calculates the 
energy of each, and then determines the energy of a super- 
position admitting resonance among them. 

The valence-bond theory has much room for improvement. 
Seeing how H 2 is improved is a guide to seeing how other 
molecules are improved. The flaw in the simple picture of H 2 
lies in the method's implicit insistence that, if an electron 
occupies an orbital on one atom, then the other electron must 
be on the other atom. In practice we know that there is a 
significant possibility that both electrons will be found on the 
same atom, and so the wavefunction ought to be improved by 
permitting an admixture of ionic contributions a\a 2 , corre- 
sponding to H~H t ,and/) 1 62, corresponding to H + H~. According 



to the "variation principle we know that an improvement of 
the wavefunction leads to a lower energy, and this is found 
when an ionic component is allowed to contaminate the 
covalent wavefunction. The name given to this mixing in 
valence-bond theory is ionic-covalent resonance because the 
molecule resonates between the forms and the energy relaxes 
in the normal way. 

This discussion has led us to the point where we are able to 
mention the disadvantages of the valence-bond theory, disad- 
vantages that have offset the advantage of chemical plausibility 
at the root of the theory. One disadvantage is that the number 
of canonical structures which ought to be included increases 
dramatically with the number of atoms in the molecule. For 
example, there are 5 structures for benzene, but more than 
100 000 for coronene. The structure must be allowed to 
resonate around all these canonical forms, and so it can be 
appreciated that the determination of the energy and structure 
of a moderately large molecule is a task of enormous magni- 
tude. Another difficulty is the importance of the ionic 
structures which must be added to the canonical structures. The 
importance of these increases dramatically as the number of 
atoms increases. The modern tendency, however, is to recon- 
sider these disadvantages, and present indications suggest that 
valence-bond theory is about to be given a second chance. 

Questions. 1 . What is the inspiration for the valence-bond 
approach? What is the perfect-pairing approximation? Why 
is it only an approximation? What happens when there are 
several perfect-paired structures of similar energy? What is 
the consequence for the energy in that case? Give an 
example of this situation in the case of H 2 and benzene. 
Discuss the VB description of molecular hydrogen. What 
is the nature of the reduction in the energy of the molecule 
relative to the energy of the separated atoms? What is the 
role of electron exchange in Hj? Why, then, are electron 
pairs so important for the formation of the chemical bond? 
2. The VB wavefunction for H 2 is (a 162 + 82 &i ) > n tr| e 
notation used in the text. Deduce an expression for the 
energy of the molecule in terms of this wavefunction and 
the correct "hamiltonian, and confirm that the integrals 
that arise are those shown in Fig. VI and Box 29. Show that 
for the alternative combination {aib 2 —g 2 bi) the energy 



valence bond 



249 



is determined by the same integrals but with some different 
signs. Assess the sign of the integrals and thence show that the 
lower-energy wavefunction is the former. What, according to 
the 'Pauli principle, can be said about the spins in the bond? 
If the spins were parallel (unpaired) what would we be forced 



BOX 30: The VB secular determinant 

1. Form all the canonical structures within the perfect- 
pairing approximation (for example, the two Kekule and 
three Dewar structures of benzene). If then desperate with 
complexity, select a manageable number of important 
structures on the grounds of chemical intuition (for 
example the two Kekule structures). 

2. Superimpose each structure with itself and all other 

structures and calculate the element H — ES , which 

re re' 

stands in the rth row and cth column of the secular 
determinant, by applying the formula 



H. 



ES fc = fy N -'[J-E + AK] 



2/V is the total number of electrons, / is the number of 
islands formed in the superposition, and A is the number of 
connected pairs of neighbours in these islands, less half the 
number of pairs of neighbours on separate islands. We mean 
by neighbours the orbitals between which there is an inter- 
action (and typically geometrical neighbours), J and K are 
the Coulomb and exchange integrals. 

An example. Let r correspond to one Kekule form, and 
c to a Dewar form of benzene 



r 1 + 1 H 




A/ = 3,4=3+1-j2 = 3,/ = 2 

H rc~ £S rc = ^- £ + ^ 
3. Construct the full secular determinant (5 X 5 in the 
case of benzene), find the roots E, select the lowest root as 
the energy of the molecule, and find the coefficients in the 
superposition that corresponds to this root. The squares of 
the coefficients give the weights of the canonical structures. 



to conclude? The variation of the molecular integrals with 
distance is given in Fig. V2 for the case of H 2 . From the 
values, deduce the energy and the bond length. 
3. What is meant by a 'canonical structure'? How many 
canonical structures are there for benzene? One way of 
deducing the number of canonical structures is by means of 
the Burner diagram, where all the contributing orbitals are drawn 
on a circle and then pairs are joined until there are no unpaired 
points. The number of structures that can be drawn in this way 
without any lines crossing is the number of canonical structures 
for the problem. Investigate this device for benzene and 
naphthalene. Can you deduce a general formula for the number 
of canonical structures for aromatic molecules? The way that 
the energy of each structure and the energy of interaction 
between structures is calculated is set out in Box 30. Each 
canonical structure is superimposed on each other (and itself) 
and the energy is related to the number of 'islands' formed by 
the superimposed tines. Use the formula in the Box to deduce 
the energy and interaction energy of the two Kekule structures 
of benzene, set up the "secular determinant in order to 
determine the energy and structure of the best (lowest-energy) 
superposition, and deduce that it consists of 50 per cent of each 
structure with an energy J + 2-4/C, and therefore that the 
"resonance energy is 0'9 K. Now include the three Dewar 
structures, and express the state of the molecule as 

C K^ K J + ^' +C D^ D) + ^D2 + ^D3l' inan0bUiOUS 

notation (I hope); set up the appropriate secular determinant, 
and deduce the extra stabilization energy that arises from 
admitting the Dewar structures. For what proportion of the 
structure do they account? 

Further information. See MQM Chapter 9. A nice account of 
simple VB theory, and an extensive comparison with molecular- 
orbital theory is given in Chapter V of Coulson (1961 ) and by 
Murrell, Kettle, and Tedder (1965). See Eyring, Walter, and 
Kimball (1944) for details of the method. Pauling's classic 
book (1960) is almost exclusively an account of the VB 
description of molecules and in its earlier editions is an un- 
paralleled example of the power of quantum- chemical reason- 
ing within the format of the theory. The molecular-orbital 
and valence-bond theories are discussed in comparison under 
"molecular orbital versus valence bond. 



250 



valence state 



valence state. Let us centre our attention on carbon, and in 
particular on the carbon atom in methane. A chemical descrip- 
tion of the structure of methane might regard it as an 
sp 3 -"hybridized carbon with each of its four tetrahedral lobes 
overlapping one of the four surrounding hydrogen atoms. 
Therefore the bonds are formed from four hydrogen atoms 
overlapping the four tetrahedral orbitals of a carbon atom in 
its valence state 1s 2 2s2p v 2p i/ 2p The valence state is the 

X y £ 

state of the atom responsible for its bonding to its neighbours. 
From this definition it is a trivial consequence that the valence 
state of the hydrogen atom is simply 1s; similarly the valence 
state of the oxygen atom is 1s 2 2s 2 2p^2p, 2$^ in a substance 
such as water. 

It is important to realize that the valence state is not in 
general a spectroscopic state; that is, it cannot be detected by 
lines in a spectrum representing transitions into or out of it. 
The clearest way of appreciating this is to reflect on the nature 
of the bonding between the carbon and the hydrogen atoms 
in methane, and to recognize that although the electrons are 
"paired in individual bonds there is no pairing of spins in 
different bonds. Therefore the valence state is characterized by 
the four valence electrons with random relative spins. But a 
state with random spins is not a spectroscopic state, for in 
these there is a strict coupling of the various angular momenta, 
and therefore a strict distribution of relative spin orientations. 

It is of interest to be able to know the energy of the valence 
state, especially when the importance of "hybridization is being 
assessed; this may still be done even though the state is not 
spectroscopic. The valence state may be expressed as a mixture 
{'superposition) of true spectroscopic states, and its energy 
calculated by the corresponding average of the energies of the 
contributing states. In this way it is possible to assess the energy 
required to promote an electron from the ground state of 
carbon to form the valence state {7 eV, 680 kJ mol"' ). 

Is the valence state ever formed? Since it is not a spectros- 
copic state the answer is strictly no (but see Questions); but we 
may envisage the valence state as emerging from the ground 
state as the ligand atoms are brought up towards it. Once again 
consider methane, but a 'potential methane' in which the four 
hydrogen atoms are disposed tetrahedrally at infinite separation, 
and a central ground -state carbon atom. As the atoms approach 
tetrahedrally the surface of the carbon atom begins to stir, and 



for brief moments the electron density might tend to accumu- 
late tetrahedrally. As the atoms get even closer the fluctuations 
are stronger and more long-lasting; and when the atoms are at 
their equilibrium bonding distance the fluctuations are massive 
and essentially frozen, forming the four tetrahedral a-bonds. 
Only at this point would it be true to say that the central atom 
was in its valence state, which has been drawn out of the ground 
state by the presence of the hydrogen atoms and the bond 
energy that lowers the energy of the whole system. 

Questions. What is a valence state? Why is it not a spectroscopic 
state? How may the energy of the valence state be determined? 
What role does the valence state play in chemical -bonding 
theory, and why is its energy important? The valence state can 
be expressed as a superposition of spectroscopic states; suppose 
we contrived to produce an atom in a state which was just such 
a superposition of spectroscopic states, discuss the history of 
the state from its moment of formation. 

Further information. See MQM Chapter 9. See also Coulson's 
The shape and structure of molecules (OCS 9) and §8.4 of 
Coulson (1961 ), who gives the following references to 
calculations on valence states, their composition, and their 
energy: Voge (1936), Mulliken (1938), Pauling (1949), Moffitt 
(1950), and Skinner (1953, 1955). A helpful discussion of 
valence states, with examples, is given in §4.1 of McGlynn, 
Vanquickenborne, Kinoshita, and Carroll (1972). 

variation theory, or variation principle. The energy 
calculated using an arbitrary "wavef unction cannot be less 
than the true lowest energy of the system. 

In quantum theory we are told to calculate the energy of a 
system by evaluating the "expectation value of the "hamiltonian 
of the system, and so we evaluate the quantity E= 
JdT ^W^/Jdr^*^. The variation principle informs us that 
if we make an arbitrary choice of the function <f> then the 
analogous quantity 8= fdT<p*HigffdT<p*<p, which is called the 
Rayleigh ratio, cannot be less than the true ground-state energy. 
The implication of this important result is that if we make a 
series of guesses about the form of the trial function, the one 
that gives the lowest energy will most closely resemble the true 
wavefunction of the system. If we are lucky we shall guess a 
function that yields the true energy: in that case we shall have 



vector model of the atom 



251 



found the true ground-state wavefunction. It is important to 
develop a method of choosing the best function other than 
relying on mere intuition, and two techniques are often 
employed. 

The first writes the wavefunction as a function of one or 
more parameters, and then varies the parameters in search of 
a minimum. Thus if the trial function were dependent on the 
value of a parameter p, differentiation of £{p) with respect to 
p and determination of the condition for a minimum yields the 
best value of p and therefore the best function of that particular 
form. Of course we might have chosen a function of a poor 
form, but the function so found would be the best of that type. 
As an example, we might have guessed that the ground state of 
the "hydrogen atom was welt described by a function exp{— pr 2 ): 
a variation treatment leads to a best value of p, but not a very 
good energy. A function exp(— pr) requires another value of p, 
and in this case the best energy is the exact ground-state energy, 
and therefore the trial function with this value of p is the exact 
ground-state wavefunction. 

A different approach was introduced by Ritz: he supposed 
that the trial function could be written as a sum of functions: 
the functions themselves are invariable, but the amounts of 
each in the mixture constitute the variable parameters. The 
trial function is of aformv3=Pi ^i +p 2 2 + , . . ,and the 
minimum is found by differentiating &(p t ,p 2 , , . . ) with 
respect to all the parameters p f and seeking a simultaneous 
minimum. This procedure is the basis of the method of deter- 
mining the best mixture in the method of "linear combination 
of atomic orbitals. Once again the minimum energy differs 
from the true energy if an insufficiently flexible trial function 
has been chosen: it also differs if an approximate hamiltonian 
has been used (in which case an energy below the true energy 
may be found, for the hamiltonian must be correct if the 
variation method is to be tried). 

It would be very useful to know how far the variational 
minimum energy, which is an upper bound to the true energy, 
lay above the true energy. There are techniques of finding a 
lower bound (beneath which the true energy cannot lie), and 
in principle this gives some indication of the accuracy of a 
variational calculation; but the technique is difficult and has 
not been widely used. 



Questions. 1. State the variation principle. Does it provide an 
upper or a lower bound to the true energy? If one guesses a 
wavefunction and calculates the energy of the system with it, 
of what can one be sure? What are the two methods of 
selecting the best trial function of a particular form? Why 
might the energy so calculated still be considerably greater 
than the true energy of the system? What is the Ritz procedure? 
What should be determined in order to estimate the error in the 
variation calculation? 

2. Take a trial function of the form exp(— pr) and vary p to 
find the ground state of the hydrogen atom. (The form of 
the "hamiltonian will be found in Box 7 on p. 90 and the 
radial part of the 'laplacian in Box 11 on p. 124). Do not 
forget to maintain the normalization of the function; in 
other words, minimize the ratio fdTip*H$fdT<p*tp with 
respect top. Repeat the exercise with a trial function of 
the form exp(— pr 2 ). Now try a function of the form 
piexp(— pit 2 ) +p 2 exp(— piT 2 ) and attempt to achieve a 
lower minimum. Sketch the form of the three best trial 
functions. 

3. The Ritz procedure takes a trial function of the form 
?p.i/<. and varies the parameters p.. Show that the condition 

fit I 

for a minimum energy is attained when the determinant 
I W-- - SS..| vanishes. H.. are the integrals fdT^/^H^f- and 5 ( .. 
the integrals fdrify. *^-. The minimum energy is the smallest 
root of the determinant, and the determinant itself is known 
as the -secular determinant. Apply the Ritz variation principle 
to the demonstration that the Is-orbitals in molecular hydrogen 
contribute equally to the bonding molecular orbital. 

Further information. For a simple proof of the variation 
principle and a derivation of the minimum conditions, see 
MQM Chapter 7, and §3.6 of Coulson (1961). For further 
discussion see §7c of Eyring, Walter, and Kimball (1944), 
Kauzmann (1957), Pilar (1968), and Wilcox (1966). For a 
discussion of the determination of lower bounds see Lowdin 
(1966) and references therein. 

vector model of the atom. The vector model is a represen- 
tation in terms of vectors of the coupling of the angular 
momenta of the electrons of the atom. The basis of the method 



vector model of the atom 



is the representation of a state of "angular momentum of 
magnitude [/, (/, + 1(j *h by a vector j, of length [/, (/, + 1 )] * 
with an appropriate orientation. If the component of 
angular momentum on some arbitrary z-axis is well defined 
and has the value mh the orientation of the vector is drawn so 
that its component on this axis is of length m. Since such an 
angular momentum "processes about the 2-axis, the vector ts 
drawn so that it lies on a cone at some arbitrary but indeter- 
minate azimuth (Fig. V3). 

a 



lenqrh /JCJ4-D' 




classical trajectory 

FIG. V3. The basic vector model of angular momentum. 

If a second source / 2 is present, the total angular momentum 
of the system may be constructed as the resultant of the two 
vectors representing the two momenta: since the length of the 
resultant vector must be [/(/ + 1 )] w , with the value of/ selected 
from the set/, +/ 2 ,/, + /i - 1 , . . . I/, -j 2 \ (see -angular 
momentum), only a few orientations of the three vectors 
ii.h. and their resultant j are permitted. In accord with the 
algebraic theory of angular momentum the vectors j i and j 2 
precess around their resultant, and the latter precesses around 
some arbitrary z-axis. This situation is represented by a vector 
diagram of the type shown in Figs. V4 and V5. The process of 
coupling the momenta together may continue until all the 
individual spin and orbital contributions have been combined 
into the one resultant representing the total angular momentum 
of the whole atom. Fortunately this formidable exercise can be 
simplified by a number of approximations. The first simplifi- 
cation is to note that the core of the atom (the electrons in 
other than the valence shells) has zero angular momentum 
because all its spins are paired and the shell is complete. The 




FIG. V4. Coupling of j i and jj in the vector model. 




FIG. V5. Coupling of j t =2 and/ 2 - 1 to give/ = 3, 2, 1. A vector 
construction is shown in black, and a simple rule of thumb, which 
uses lines of length Jt.hJ. is shown in colour. 

next approximation supposes that there are two extreme cases 
of coupling. 

1 . The first, the Russell-Saunders coupling case or LS-coupl/ng 
case, assumes that the °spin-orbit coupling is so small that it is 
effective only after all the orbital momenta have been summed 
into some resultant L, and all the spins summed into another 
resultant S. The electronic orbital motions are dominated by 



{ 



vector model of the atom 



253 



the electron-electron electrostatic interactions, and this is the 
source of their coupling energy. For two electrons the coupling 
of I j and [ 2 would be represented by a diagram of the type in 
Fig, V6 a. The two spins also couple to give a definite resultant 
S; the coupling energy for this interaction arises from the 
spin -dependent 'exchange energy, and so it too is an electro- 
static phenomenon. At this stage the two resultant momenta 
L and S couple together to form a resultant J, the total angular 




elecrrastaric coupling 
S. 




-• 



Spin-orbif coupling 





clecrrosforic coupling 



FIG. VG. (a) Russell-Saunders or LS-coupling. (b) //-coupling. 



momentum, and the strength of this interaction depends on 
the strength of the spin-orbit coupling. 
2. When the spin-orbit coupling is stronger than the electro- 
static interaction the Russell -Saunders scheme breaks down 
because the spin and orbital angular momenta of individual 
electrons attempt to organize themselves into satisfactory 
mutual orientations. The jf -coupling scheme describes the 
extreme situation of this kind. In it each electron's spin is 
allowed to couple with its orbital momentum; thus l t and 
Sj couple to form Ji . The two components "process strongly 
around their resultant in the manner characteristic of a 
strongly interacting pair of momenta. This jj is now coupled 
to another j 2 , and the total angular momentum J constructed: 
the latter coupling is relatively weak, for it depends on the 
electrostatic interactions of the electronic orbitals. We see 
that, although the total angular momentum obtained in this 
way might be the same as the total in the Russell -Saunders 
scheme, the states of the atoms are different, and their 
energies also differ. 

Neither scheme is an exact representation of the true state 
of affairs because there is always some competition between 
the different types of interaction, and indeed it is quite 
possible for some electrons in the same atom to be coupled by 
one scheme and the remainder by the other. Nevertheless 
for light atoms, which have small spin-orbit coupling 
constants, the Russell-Saunders scheme is often a good 
description of the valence electrons. Heavy atoms, which have 
large spin -orbit coupling constants, are often predominantly 
//-coupled. It follows that the wavef unctions corresponding to 
the Russell -Saunders scheme are a good starting point for 
more elaborate calculations on light atoms. 

The angular momenta that one is led to by the vector 
model are the bases of the labelling of the state of atoms by 
"term symbols. 

Questions. What is the vector model of the atom? What are 
its basic features? What is the length of the vector that would 
represent the orbital angular momentum of an eiectron in an 
s-orbital, a p-orbital, and a d-orbilal? What is the length of the 
vector representing the "spin of an electron? Construct vector 
diagrams for the coupling of the spin and orbital angular 
momenta of an electron in a p-orbital. What is the energy of 



254 



vibrational spectroscopy: a synopsis 



interaction? What approximations may be introduced to 
simplify the discussion of the coupling of angular momenta in 
atoms? By a vector construction show that the angular 
momentum of a complete K-shell (Is 2 Ms zero. What is the 
source of the coupling energy between the orbital momenta 
of electrons? What is the source of the spin coupling energy? 
When is it appropriate to use Russell -Saunders coupling? When 
should //-coupling be used instead? Are there any alternatives? 
Is it possible to use the Russell-Saunders term symbols even 
when the//-coupling predominates? 

Further information. See MQM Chapters 6 and 8 for a further 
discussion of the vector model of the atom and the two 
coupling schemes. Good accounts are given in §1 1 .2 of 
Herzberg (1944), Chapter V of White (1934), §3.4 of King 
(1964), and Chapter 12 of Kuhn (1962). Candler's book 
(1964) concentrates on the vector-model description of atoms. 
"Hie molecular situation is outlined in the section on the -Hund 
coupling schemes. 

vibrational spectroscopy: a synopsis. As a first approxi- 
mation the vibrations of molecules are assumed to be simple 
•harmonic. The frequency depends on both the "force-constant 
and the mass of the vibrating object according to o>= (it/m) H 
in radians per second. Molecular force-constants and masses are 
such that vibrational frequencies fall in the infrared region of 
the spectrum, and so obtaining a vibration spectrum is an infrared 
spectroscopic technique. Vibrational structure also appears in 
an "electronic spectrum, for during an electronic transition 
vibrations may be excited: the spacing of the lines of this 
vibrational structure is of the order of an infrared frequency. 
As a rough guide, weak bonds between heavy atoms vibrate in 
the region of several hundred cm" 1 {the I -I bond in \ 2 vibrates 
at 214 cm" 1 , 6-4 X 10 11 Hz, or 4-67 X 10 4 nm, in units of 
wave number, circular frequency, and wavelength respectively), 
and stiff bonds between light atoms vibrate in the region of 
several thousand cm" 1 (the H-H bond in hydrogen vibrates at 
4395 cm" 1 , or 1-3 X 10 w Hz, or 2280 nm). Bond stretches 
tend to be at higher frequency than bond bends. See Table 10 
for the vibrational frequencies of some diatomic molecules, 
and Table 24 for the typical frequencies of groups in molecules. 



A vibration is active-i.e. observable -in the infrared if 
during it the dipole moment of the molecule changes. This 
implies that diatomic molecules absorb in the infrared only 
if they are polar. In more complicated molecules it is 
necessary to scrutinize the form of the 'normal mode in 
order to see whether the vibrations of several atoms 
jointly lead to an oscillating dipole. This may often be done 
by inspection, but more positively one should take into 
account the symmetry of the system by using "group theory. 
If the vibration is active the selection rule for the transition is 
that the vibrational quantum number for that transition may 
change by ± 1 . At normal temperatures the Boltzmann 
distribution ensures that essentially all molecules are in their 
ground vibrational state: this implies that the spectrum should 
consist of a single line for each vibrational mode of the 
molecule, and correspond to the excitation of the mode of 
vibration from its ground state to its first excited state. Such 
a line is the fundamental and is denoted (I* — 0). 

The vibration of a molecule is not strictly harmonic because 
the potential in which the atoms move is not strictly parabolic: 
the deviations are greater at large displacements from equilib- 
rium. This 'anharmonicity has several consequences. First, the 
selection rule for the transition fails to a degree that depends 
on the amount of anharmonicity. Instead of a single line for 
each mode one sees the fundamental <1<— 0), or the first 
harmonic, accompanied by weaker overtones, or second-, 
third-, . . . harmonics corresponding to the 'progression of 
transitions (2*— 0), (3* — 0), . . .. These should appear at the 
frequencies 2w, 3cj, ... but not exactly because of the 
anharmonicity. It is possible for two modes of a molecule to 
be excited simultaneously if they are not entirely independent 
(that is, if there is present anharmonicity which is able to mix 
together the two modes) : the absorption that is responsible for 
this appears as a combination band. When the energy of a 
combination level lies close to the energy of an unex cited 
mode, which may be unexcited because it is inactive in the 
infrared, there may occur a 'resonance interaction between 
them by virtue of the anharmonicity present. This Fermi 
resonance causes the lines to shift, and the active bands donate 
intensity to the inactive, which therefore appear in the spec- 
trum (this is intensity borrowing, brought about by the inactive 



vibronic transition 



255 



mode acquiring some of the properties of the active modes : 
see "superposition principle). 

When a vibrational transition occurs it may be accompanied 
by a "rotational transition of the molecule. This gives rise to a 
structure in the spectrum which is observable when the sample 
is gaseous; in a liquid collisions with the solvent blur the 
structure by reducing the lifetime of the rotational states. In 
an -electric dipole transition the "rotational "quantum number 
J may change by or ± 1 ; consequently there are lines at the 
position of the pure vibrational transition (A/= 0) which 
constitute the Q-'branch of the spectrum; a series of lines at 
lower frequency (/\J = -1 ), the P-branch, and a series at higher 
frequency (A J = + 1 ), the R -branch. A series of lines rather 
than a single line is observed because the Boltzmann distri- 
bution permits a number of rotationai levels to be occupied 
in the initial state, and each one of these gives rise to a line 
in the branch. The branches may pass through a head if the 
rotational constant of the upper vibrational level is signifi- 
cantly different from the rotational constant in the lower 
level (see "branch); this is especially important when the 
vibrational transition is part of an electronic transition for then 
the rotational constants may be very different. 

The other features that affect a vibrational spectrum in- 
clude "inversion doubling (for example, in NH 3 ) and t-type 
"doubling in linear tri atomic molecules. See "Coriolis inter- 
action. 

The main pieces of chemical information that one may 
obtain from a study of vibrational spectra include the elemen- 
tary but important one of the identification of a species by 
using its vibrational spectrum as a fingerprint. The major quan- 
titative information that may be obtained is the rigidity of 
bonds under the stresses of stretching and bending: the 
force-constant is an important feature of a chemical bond. The 
anharmonicities show how far the true potential differs from 
an ideal parabola. The rotational structure on vibrational 
transitions enables the molecular geometry to be determined 
in different vibrational states (bond-angle and bond-length 
dependence on vibrational state), and the vibrational and 
rotational structure of electronic transitions enables the same 
kind of information to be obtained about electronically 
excited states. This information enables one to build up a 



full picture of the potential -energy curves of molecules in 
different electronic states. 

We have concentrated on electric dipole absorption spectra: 
vibrational transitions may also be observed in "Raman 
spectroscopy. 

Further information. See MQM Chapter 10 for a discussion of 
vibrational and rovibrational spectra in more quantitative 
terms, and with the use of group theory. An introduction to 
the vibration of molecules may be found in Barrow (1962), 
Whiffen (1972), and King (1964). The characteristic frequencies 
of many bonds are listed in Bellamy (1958, 1968) who also 
describes infrared spectroscopy as an analytical tool. More 
advanced discussions are given by Gans (1971), Herzberg 
(1945), Wilson, Decius, and Cross (1955), and Allen and Cross 
(1963). See also Gaydon (1968). 

vibronic transition. The word 'vibronic' is an amalgam of 
w'6ration and electrowc, and implies that the transition involves 
both modes of excitation simultaneously, A vibronic state is 
the name applied to a state of the molecule when it is improper 
to view the electronic and the vibrational states as independent. 
In order to elucidate this description consider an octahedral 
complex and an electronic transition of a d-electron. In the 
"crystal-field or "ligand-fiefd theories of transition -metal 
complexes the d-orbitals are split into two groups sep- 
arated by a small energy difference; therefore it is tempting 
to ascribe the colours of transition-metal complexes to a 
transition of an electron from one set of d-orbitals to the other. 
The problem that immediately confronts us is the Laporte 
"selection rule, which forbids d-d transitions because it forbids 
"gerade-gerade (g-g) transitions. Most rules can be evaded, and 
one of the rules for looking for ways of evading rules is to 
seek the approximation on which that rule might be based, 
and then to repair the approximation. The Laporte selection 
rule is based on the existence in the complex of a centre of 
symmetry, and only if the complex is strictly octahedral is the 
rule strictly valid. But the complex may vibrate, and some of 
the vibrational modes destroy the centrosymmetric nature of 
the molecule. Now consider the unexcited, vibrationally 
quiescent molecule, and a photon approaching it. Let the 



256 



virial theorem 



photon excite simultaneously a d-electron and a vibration of 
the complex. Then if the excited vibration is one that destroys 
the centrosymmetry of the complex, the Laporte rule will 
be slightly but sufficiently broken, because the complex no 
longer possesses a centre of inversion in the initial and final 
states. 

The transition is allowed only in so far as it is proper to 
treat the vibration and the electronic motions as coupled 
together so that they jointly determine the symmetry of the 
object with which the light is interacting. Therefore the 
transition is vibronic and the states of the complex must be 
treated as vibronic states. This view leads to another way of 
looking at the nature of a vibronic transition. In this we 
consider the possibility of a transition from a d -orbital to a 
p-orbital; this, being a g to u transition, is allowed. But why 
should the upper level be a p-orbital, or at least why should it 
possess some p-character? If the electrons follow the nuclear 
vibrations the electronic distribution in the upper state must 
follow the nuclear motion. To do so it must distort from the 
eentrosymmetric distribution which d-orbitals give rise to, 
and one way of achieving the distorted distribution is to mix 
in ('hybridize) some p-orbital character (Fig. V7). Therefore, 
when such an antisymmetric mode is excited the electronic 
distribution contains some p-character; and as d-p transitions 
are allowed, the transition from the quiescent ground state to 
the vibronic upper state is allowed in proportion to the 
amount of p-character the vibration introduces. 

Further information. See MQM Chapter 1 for a further 
discussion, and some group theory. Vibronic transitions are 
described in more detail by Orgel (1960), Figgis (1966), 
Ballhausen (1962), and Griffith (1964). When the rotational 
states of a molecule must be considered too then a 



rovibronic transition occurs, and the composite states are 
rovibronic states. 



virial theorem. In its simplest form the virial theorem states 
that if the potential energy of a system follows a Mr law then 
the mean kinetic energy Is related to the mean potential energy 
by the expression < 7"> = — ~<V>. This in turn implies that 
the total mean energy of the system is simply equal to ^<.V>, 
or to — <T>. A more general form of the theorem states that 
if the potential energy follows the tawf", then the mean 
potential energy and the mean kinetic energy are related by 
<T> = ^n<V>. Thus for an harmonic oscillator, in which 
the potential energy is parabolic (r> = + 2), the mean potential 
and kinetic energies are related by < T> — < V>, and this 
equality is yet another manifestation of the peculiarly high 
degree of symmetry of the 'harmonic oscillator. The theorem 
applies to a bounded system, and one that is stationary in 
time; but if by mean value is also implied a time average, then 
it also applies to no n -stationary states. 

The name 'virial' (which is derived from the Latin vis, vires; 
force, forces) comes from the classical mechanical form of 
the rate of change of the mean value of the product r.p, where 
r is a position coordinate and p is the linear momentum. This 
leads to the equation < T> = - j< F.r>, the virial theorem of 
Clausius, where F is the force acting. From this expression may 
be deduced the general form of the equation of state of a 
real gas in which forces operate between the gas particles: this 
gives rise to the virial expansion and the virial coefficients of 
thermodynamics. The virial theorem may also be derived 
quantum mechanically and applied to a discussion of the 
structure and properties of atoms and molecules. For example, 
it is a test of the exactness of a calculated wavefunction that 




FiG. V7. An asymmetric vi- 
bration hybridizes p and d i and 
permits a d-d transition via the d-p 
component. 



virtual transition 



257 



the expectation values of the potential- and kinetic-energy 
operators do indeed satisfy the virial theorem. It is essential to 
remember that the virial theorem imposes a connexion between 
the way that the kinetic and potential energies vary as wave- 
functions are distorted. The virial theorem may also be used as 
an alternative to the -variation theorem in some circumstances. 
A further generalization of the virial theorem may be made 
and the hypervirial theorems obtained. These are a set of 
theorems based on the vanishing of the average value of the 
"commutator of an -operator with the -harniitonian of the 
system when the system is in an -eigenstate of the hamiltonian. 

Questions. From the wavef unctions given in Tables 1 1 and 1 5 
demonstrate the validity of the virial theorem for the ground 
states of the harmonic oscillator and the hydrogen atom. 
Return to the calculation of the ground state of the hydrogen 
atom in terms of the -variation principle, and investigate 
whether the viria! theorem is satisfied for the best gaussian 
approximation. 

Further information . For a good discussion of the classical 
virial theorem, its deduction, and some applications, see 
Goldstein (1950); for the deduction of the gas laws see 
Lindsey (1941). The quantum-mechanical virial theorem is 
derived by Hirschfelder [1960), who also deduces and 
describes the hypervirial theorems. Application of the 
theorem to molecular- and atomic-structure calculations are 
described by Coulson (1965), Deb (1973), and Feinberg, 
Ruedenberg, and Mehler (1970). 

virtual transition. When a system is 'perturbed, for 
example by the application of an electric field, it is distorted; 
the distorted system can be described by a -superposition of 
the -wavef unctions of the states of the original system, and 



therefore the system behaves as though it contained features 
of the excited states; it has made a virtual transition to the 
excited state. When a light wave scatters from a molecule the 
distortion it induces can be envisaged as a series of virtual 
transitions to the excited molecular states caused by a virtual 
absorption of a photon. The distortion is immediately released, 
and the photon flies off leaving the molecule either in its 
original state (Rayleigh scattering) or in one of the excited 
states populated by the initial virtual transition (-Raman 
scattering). 

As the frequency of the incident light approaches one of 
the transition frequencies of the molecule the transition 
gradually loses its virtual character and becomes real: the 
molecule is really excited and the photon is really absorbed. 
Energy is conserved in real transitions; but as 'virtual transition' 
is just a name for a way of describing a distortion and of taking 
into account the effect of a perturbation, for them it is not 
conserved. 

Questions, What is a virtual transition? In what ways does 
it differ from a real transition? When does a virtual transition 
take on the character of a real transition? Is energy conserved 
in a virtual transition? What virtual transitions are involved 
when an electric field is applied to a "hydrogen atom? 

Further information. See MQM Chapter 7 for a discussion of 
perturbation theory and a concomitant discussion of virtual 
transitions. Books that deal with "perturbation theory perforce 
deal with virtual transitions, although the term is not always 
used. See Davydov (1965), Dirac (1958), Landau and Lifshitz 
(1958a), Messiah (1961), and Schiff (1968). See §16 of Heitler 
(1954) and §7.5 of Hameka (1965) for a discussion of the 
transformation of virtual into real transitions. See -Raman 
effect and -Stark effect. 



w 



wavef unction. The wavefunction for a system is a solution of 
its Schrodinger equation and is the function that contains all 
the information about its dynamical properties. If the wave- 
function that describes the state of the system is known, all the 
observable properties of the system in that state may be 
deduced by performing the appropriate mathematical 
"operation. The wavefunction may be a function of time, and 
is then often written ^(r,t). When it is not a function of time 
(or when the time-dependence has been factored out) it is 
often denoted i^, and is a function of all the coordinates of 
all particles that make up the system. Since the wavefunction 
depends on the state of the system it is often labelled with an 
index or set of indices {the "quantum numbers) that distinguish 
the state. Thus the wavefunction for a system containing N 
particles and needing M quantum numbers is the mathematical 
function ty n (r! ,r 2 . . . r w ). As examples of wave- 

functions we may point to the wavefunction for a free particle 
travelling in the x -direction with a 'momentum kh, which is 
the function exp \kx, and to the wavefunction for the ground 
state of the "hydrogen atom, which is the simple function 
exp(— r/a ). The wavefunction has an interpretation, must obey 
some restrictions, and contains information. We describe these 
aspects below. 

1. Interpretation of the wavefunction. We concentrate on a 
system containing one particle with the coordinate x. The 
Born interpretation of $W is that it is the amplitude for 
the probability distribution of the position of the particle. 
According to this interpretation the probability of finding 
the particle in the infinitesimal range dx surrounding the 



point* is proportional to i^*(x)^(x)dx. The probability 
density at the point x is therefore proportional to 
i/<*(x)i//(x). If we were dealing with a three-dimensional 
system the wavefunction t//(r) would be interpreted as follows: 
i//*(r) \jj[f)dT is the probability of finding the particle in an 
infinitesimal volume element tiT surrounding the point r. The 
interpretation may be pictured in terms of inserting a probe 
sensitive to the presence of the particle, and which samples a 
volume dT in the system; as the probe is moved to different 
points r the meter reading is proportional to the volume of 
the probe and to the value of \jj*{r)\p{r). As an example, the 
wavefunction for the "hydrogen-atom ground state is a 
decaying exponential function of r; therefore the meter reading 
for the electron density will fall according to exp(— 2r/a )dT as 
the probe of volume dT is moved out along a radius. In the case 
of the other wavefunction referred to above (expifcx) the meter 
would give the same reading wherever the probe is inserted 
because {expifcx)*{exptfrx) = 1 and is independent of x. This 
function corresponds to an even spreading of the particle 
throughout the universe, whereas for the hydrogen atom the 
electron is densest close to the nucleus. 

2. Limitations on the wavefunction. If the wavefunction is to 
be interpreted as an amplitude for the probability density for 
the distribution of the particle is must be constrained in a 
variety of ways. 

(a) It must be finite everywhere, for otherwise there would 
be an indefinite accumulation of probability density at the 
points where it became infinite. (This requirement is really too 
stringent: all we need to impose is the condition that the total 



258 



wavef unction 



259 



probability of the particle being within the universe is unity— 
thus we require the existence of the integral fdr\j/* (r)\p(i); but 
our too-stringent requirement is a good guide in most cases.) 

(b) The probability density must be single-valued every- 
where, because it would be nonsense to say that the prob- 
ability density at a certain point is both 02 and 04, In most 
cases {systems involving "spin are exceptions which are easily 
accommodated in another way) this requirement is the same 
as requiring the wavef unction itself to be single-valued. 

(c) The wavefunction must be continuous, for it would be 
unreasonable to have a probability density of a particular 
value at a point and a finitely different value an infinitesimal 
distance away. 

The imposition of these limitations on the wavefunction 
is severe, for it forces it to obey certain boundary conditions, 
and leads ineluctably to quantization (see "quantum) because 
only a very few of the solutions of the -Schrd'dinger equation 
survive when the conditions are imposed. 

3. Information in the wavefunction. We have already seen that 
the wavefunction contains the information about the distri- 
bution of the particle. The mean gradient (slope, first 
derivative) of the function is the 'momentum of the particle 
in that state; this emerges from the quantum-mechanical 
rules about interpreting observables by 'operators. The mean 
curvature (the second derivative) is the -kinetic energy of the 
state. The value of any observable is determined by calculating 
the "expectation value of the corresponding "operator using 
the appropriate wavefunction. 

4. Time-dependent wavef unctions. If the wavefunction 

^„ (x) corresponds to an energy ( -eigenvalue) E the time- 
dependent form of the wavefunction ^ {x,t) is simply the 
product 4> n (x)exp(-if n t/fi). This is a stationary state (even 
though t occurs) because the probability density [^ (x,t)\ 2 
is independent of time. 

5. Pure states and superpositions. If it is certain that a 
system is in a state with well-defined quantum numbers then 
the wavefunction is that of a pure state. As an example, a 
hydrogen atom known to be in its ground state is in a pure 
state, and a particle with precisely defined momentum is also 
in a pure state and is described by a simple wavefunction. 
When the state of a system is believed to be one of a range 
of pure states, for example, if the particle has a momentum 



somewhere in the range (k — K)n to {k + K)h, then the wave- 
function for the system is a -superposition of the pure-state 
functions covering this range. Thus if the state is believed to 
be in the range of states spanned by the functions 

^ n M< ^„ W- ■ ■ ■« tne true state o f the system is described 
i "2 

by the linear superposition 0(x) = c,0 {x)+c 2 \p [x)+ . . ., 

where the coefficients determine the probability that the 
system is in one of the basis states: the probability that the 
system is in a state described by the wavefunction \p (x) is 
proportional to c*c n , or jc p . The coefficients may be 
time-dependent. An example would be the excitation of a 
hydrogen atom by incident radiation: initially the atom is in 
the ground state described by the function $ (f), but as 
irradiation continues it takes on more of the character of the 
2p^ -state. Therefore during irradiation its state is described by 
the function *(r,f) = c 1t W^,,W + c 2p (f)0 2p (r), with 
c 1s (0) - 1 and c 2p (0) = 0, and the probability that at a time 
f it has actually made the transition to the 2p -state is 
proportional to k, (f)|*. The calculation of the coefficients 
is a task for -perturbation theory, and the example is described 
further under -transition probability. An example of a static 
superposition wavefunction is that of a -hybrid orbital, and 
another is an -LCAO -molecular orbital. 

Questions. How is a wavefunction obtained? What is its 
interpretation? What is the difference between probability 
and probability density? Sketch the meter reading for an 
electron -sensitive probe when it is dipped into a hydrogen 
atom and pushed in towards the nucleus along a radius: 
first let the probe be a minute volume element, roughly a 
cube of volume dxdydz, and then let the probe be a spherical 
shell of area 4w 2 and thickness dr (r is the radius, and so the 
sheil gets smaller the closer it is pushed towards the nucleus: 
see -radial distribution function). What are the three 
constraints on the wavefunction? Is a wavefunction of the 
formexp(+ax) a likely candidate for a wavefunction for a free 
particle? What about the function x/|x|? A particle is 
confined to a ring and the function exp im<p is proposed for 
its wavefunction: what limitation must be put on the values 
of m? How is a time-dependent wavefunction for a stationary 
state formed? Suppose the energy E were replaced by the 



260 



wave packets 



complex quantity E— iftT; what would happen to the amplitude 
of the hitherto stationary state, and how could this be 
interpreted? How should the coefficients of a superposed 
wavefunction be interpreted? A wavefunction is written 
^ + 3 % i^ ; what proportion of s- and p-character does it 
contain? Form an sp -hybrid. 

Further information. See MQM Chapters 1 , 2, and 3 for a 
detailed discussion of the solution of the Schrodinger equation 
for a variety of systems and an account of the properties and 
significance of their wavefu net ions. For a discussion of the 
interpretation of wavef unctions see Pauling and Wilson (1935), 
Landau and Lifshitz (1958a), and Schiff (1968). For questions 
about its interpretation see Bohm (1951), Jammer (1966), and 
Ballantine (1970). We have discussed elsewhere the question of 
the 'normalization of the wavefunction, and the question of 
■orthogonality. Its interpretation as an "cigenfunction of 
the "hamiltonian is important. See also the "superposition 
principle, "atomic orbitals, and the "hydrogen atom. 

wave packets. A particle that experiment or observation 
shows to be confined to a very small region of space must 
be described by a "wavefunction that is strongly peaked 
within the region and virtually zero elsewhere. A wave- 
function corresponding to a sharply defined "momentum 
has a well-defined wavelength, and so spreads over a large 
region (actually the whole) of the system; the only way of 
attaining localization is to take a ^superposition of the 
latter functions and investigate their mutual interference. 
If the superposition has been well chosen all the construc- 
tive interference occurs at a selected point and destructive 
interference eliminates the amplitude of the wavefunction 
everywhere else (Fig. W1). The square of the sharply 
peaked function is another sharply peaked function, and 
so the probability of finding the particle differs from zero 
only in the point-like region. Thus a wave packet describes 
a localized particle (and because we have a superposition 
of a vast number of energy states the momentum is 
correspondingly indefinite). 

The wave packet also moves, because all the component 
functions are time-dependent and the point of maximum 
constructive interference moves. It is possible to show that 





FIG. W1. Formation of a wave packet: (a) single momentum state, no 
localization; (b) several states, some localization; (c) many states, good 
localization; (d) infinite number of states, perfect localization, 

when the wavef unctions are the solutions of a Schrodinger 
equation for a specified potential the motion of the wave 
packet corresponds very closely to the motion predicted 
for a classical particle in the same potential. Thus we see 
how the structure of quantum mechanics underlies the coarse 
description provided by classical mechanics. One important 
difference is that the wave packet tends to spread with time, 
but this tendency is very small for massive, slow particles. 

Further information. The formation, significance, ana 
motion of wave packets are described in MQM Chapter 3, 
and made quantitative in Appendix 3.1, A good and instruc- 
tive discussion will be found in Chapter 3 of Bohm (1951), 
and those with tough teeth should consult Gofdberger and 
Watson (1964), especially Chapter 3. 

Wigner coefficient. A Wigner coefficient, or Ctebsch-Gordon 
coefficient, or vector-coupling coefficient, is the coefficient 



work function 



261 



in the expansion of a state of coupled -angular momentum in 
terms of its uncoupled components. 

As a specific example, consider the coupling of an a- and a 
ftelectron "spin into a "singlet state. This spin-paired state may 
be expressed as a 'linear combination of the uncoupled states 
in which spin 1 has orientation a and spin 2 has orientation 
ft and vice versa: the singlet state is represented by 
(W2)[a(1)jS(2) -/3(1)a(2)].The Wigner coefficient of the 
uncoupled state &(1)j3(2) is therefore 1A/2, and of the other 
uncoupled state 0(1)3(2) is— 1//2. If we attempt to construct 
a component of the triplet state from the two uncoupled states 
we discover that taking both coefficients to be + 1// 2 would 
give (1//2)[a{1)(3{2) + 0(1 )a(2)] , which should be recognized 
as the M s = state of the triplet, Wigner coefficients enable all 
such linear combinations to be written for the coupling of 
arbitrary angular momenta into the desired resultants. 

The Wigner coefficients for the coupling of a state with 
quantum numbers /i ,m% and j 2 , m-i into one with j,m are 
written <iim l j%m 1 \jm>; a modification of this coefficient, 
which being more symmetrical is easier to handle, is known as 
a 3/symbol, 

Further information . See Brink and Satchler (1968), Rose 
(1957), and Edmonds (1957) for the properties of Wigner 



coefficients. A convenient list has been published by Heine 
(1960). A collection of 3/-symbols in a convenient numerical 
form has been prepared by Rotenberg, Bivins, Metropolis, and 
Wooten (1959). 



work function. The work function of a metal is the energy 
required to remove an electron to infinity. The analogy with 
the "ionization potential should be noticed. Metals with small 
work functions can more easily lose their electrons than metals 
with high work functions. A small list of work functions is 
shown in Table 25. 

The work function plays a role in the -photoelectric effect 
and in thermionic emission. The Schottky effect is the lowering 
of the effective work function in the presence of an applied 
electric field; this arises from the combined effect of the applied 
field and the mirror charge induced by the electron as it moves 
away from the surface of the metal. 

Further information , A very readable account of these matters 
will be found in Solymar and Walsh (1970). Comprehensive 
tables of work functions are given in §9 of the American 
Institute of Physics Handbook (Gray 1972). See also 
"photoelectric effect. 




■■■1HHBHHHBI 



X 



X-ray spectra. X-rays are electromagnetic waves of the order 
of 0-1 nm (1 A) wavelength. A principal terrestrial source is 
the bombardment of metals with high-energy electrons. The 
radiation so produced consists of two components: there is a 
continuous background of radiation on which is superimposed 
a sequence of sharp lines. The latter constitute the X-ray 
spectrum. 

The continuous component, known as Bremsstrahtung. is 
formed by the deceleration of the electrons by the metal: as 
the negatively charged electron is decelerated when it plunges 
into the metal it radiates electromagnetic radiation, and if its 
initial energy is great enough there is a significant short- 
wavelength component. 

The discrete spectral lines arise from transitions within the 
core-levels of the atoms that constitute the material: the 
incoming electron has enough energy to eject an inner-shell 
electron from the atom, either completely or into some 
unoccupied upper level; one of the remaining core-electrons 
falls into the hole left by the ejected electron, and the energy 
difference is radiated. High-energy ('hard') X-rays are formed 
when the ejected electron comes from the K-shell (n — 1): an 
electron falling from the L-shell (n --- 2] gives rise to thR 
K a -line, one falling from the M-shell (n = 3) gives the 
K.-line, and so on. Softer X-rays (longer wavelength) are 
formed when the electron is ejected from the L-shell, and 
the lines L L , etc. are formed as electrons drop from the 
M- and IM- shells. 

As a first approximation the K- radiation can be treated on 
the basis of the energies of the electron levels being hydrogen- 
like, with an effective nuclear charge of {Z — \\e, to take into 



account the single electron remaining in the Is-shell. Then 
using the mydrogen-atom energy-level formula it is an easy 
matter to deduce that the frequencies of the K-radiation are 
given by {Z~ 1) 2 ffl(1/1 2 ) - (1/n 1 )]. Similar expressions for 
other lines can be written, but they would involve different 
screening constants O in Z — o. This expression shows that the 
square root of the X-ray frequency is proportional to the 
atomic number Z: this is Moseley's law, which enabled the 
elements to be put in an unambiguous order. 

Questions. What different types of X-radiation can be 
observed when a metal is bombarded with high-energy 
electrons? What is the source of the continuous background 
radiation? Why are some sharp peaks observed? To what 
transitions do the peaks correspond? What is the source of 
K^-radiation? What is the dependence of the X-ray frequency 
on the atomic number of the atom? Why is it reasonable to 
treat the energy calculation of K-radiation in terms of a 
hydrogen-like atom with atomic number Z — 1? Calculate the 
maximum frequency of the continuum X-radiation that might 
be expected when a 1 keV, 100 keV, IMeV electron beam 
strikes a target. The K-radiation from copper has a wavelength 
of 1-541 A (154 pm) and from molybdenum 0-709 A(70-9 pm): 
compute their atomic numbers. Predict the wavelength in the 
case of aluminium. 

Further information. For an account of X-ray spectra see 
Chapter XVI of White (1934), gIVC of Kuhn (1962), §iV.2 of 
Herzberg (1944), and §13.9 of Condon and Shortley (1963). 
Tables of X*ray transition frequencies are given in $7 of Gray 
(1972). 



262 



z 



Zee man effect. The Zeeman effect is the splitting of spectral 
lines into several components by a strong magnetic field. In 
the normal Zeeman effect, which is shown by atoms without 
spin, each line is split into three. In the anomalous, but more 
common, Zeeman effect, which is shown by atoms with net 
spin, the line structure is more complicated. 

In the absence of spin the only source of "magnetic 
moment is the orbital angular momentum of the electrons; the 
applied field interacts with the orbital moment and the energy 
of the state with projection M. is changed from £ to E + 
P- B BM The 2L + 1 states of a -term with orbital angular 
momentum L are therefore no longer "degenerate but are 
arrayed in a ladder with spacing p B. For example, a "P term 
will be split into 3 evenly -spaced components, and a l D term 
into 5 components with the same splitting. The -selection rule 
for an optical transition is AM L = 0, ±1, and so all transitions 
fall into three groups. The &M — set is at the position of 
the original spectral line (see Fig. Z1), those with &M. = —1 
are displaced to low frequency, and those with AM = +1 are 
displaced an equal amount to high frequency. Closer analysis 
of both theory and experiment shows that the light emitted is 
polarized: when viewed parallel to the magnetic-field direction 
the &M L - line is absent, and the A/W,= ±1 lines are 
circularly polarized (AfW^ = — 1 is left circularly polarized, 
&M L = +t is right circularly polarized). When viewed 
perpendicular to the field the AM L = line is present and 
polarized parallel to the field (it is denoted a 7r-line}; the 
AM L — +1 lines are also plane-polarized, but perpendicular to 
the field (and denoted the cr-lines: senkrecht is German for 
perpendicular). With fields of the order of 30 kG the splitting 



is about T"8 cm" 1 , and this is easily detectable. 

When a resultant "spin is present, so that the atom is in 
some -multiplet state, it is necessary to consider the effect of 
the magnetic fieid on each of the "levels of the term: the 
2/ + 1 states A-?, of a level with total angular momentum J 
have a magnetic moment (— gjH fl\)J and therefore an energy 
£ + SjVqBMj in a magnetic field. The ^-factor takes into 
account the dependence of the magnetic moment of a state on 
the magnitudes of the contributing spin and orbital angular 



/ 


P B e 


b V 




+ 


i 








\ 














* 


'p r~\ 






^ 


V_ 






I 



field off 




field on 









cr TT CT* 
FIG, 21, The normal Zeeman effect ('d— *P). 



263 



264 



Zee man effect 



momenta: it is the Lande Rvalue. Since gj depends on S, L, 
and J, the splitting of states is different in different terms, 
and although the same selection rules apply (and the 
polarizations are the same) the transitions no longer fall into 
three neat groups. As an example, consider the transitions 
'P — ■> 'S, The ^ term has a magnetic moment arising solely 
from its spin angular momentum, and since S = 1 the field 



\ 9-VS 


I 1 


1/ 




t 




\I_ 




3 |> 4 -tf 








/ 






Wo 


X. 
























h r~ 












\ 









Held off 








field on 




1 


! 


1 





















FIG. Z2, The anomalous Zeeman effect ( 3 P- 3 S). 



separates the states into three with separation 2/U B (because 
9j = 2 when L = Q,S= \,J = \\. The 3 P term has three levels 
%, 3 P,, and 3 Pi. Since L = 1 and S = 1, g , = for J ■ 0, 
gj = | for J = 1, and f. = f for J =2. The^term is therefore 
not split by the field, and the other two levels 3 Pi and 3 Pj are both 
split by the field, the former into 3 states with splitting 
(llMgfl and the latter into 5 states with the same splitting 
(Fig. Z2). On the application of the selection rules the 
spectrum is predicted to be of the form shown in Fig. Z2, and 
the considerable complexity of the situation is apparent. Note 
that the polarization characteristics of the lines can be used to 
disentangle the spectrum. At very high fields all anomalous 
Zeeman effects become normal because the field decouples the 
angular momenta: this is the "Paschen-Back effect. 

A principal use of the Zeeman effect is the determination 
of the multiplicity of terms. The splitting of energy levels by a 
magnetic field is the basis of magnetic resonance techniques: 
see "electron spin resonance and "nuclear magnetic resonance. 

Questions. What is the Zeeman effect? Under what circum- 
stances are three lines seen? When does the anomalous effect 
appear? Account for the normal Zeeman effect. Discuss the 
Zeeman effect for the transition ' D — * ' F and construct a 
diagram of the form of Fig. Z2 to illustrate the formation of 
the spectrum. What is the polarization of the lines in the 
'O — ► l F Zeeman spectrum? What would happen to the 
polarization if the direction of the magnetic field were 
reversed? What splitting would you expect in a 10 kG 
magnetic field? Why does the anomalous effect depend on 
the presence of spin? (Look at '^-value to convince yourself 
that the magnetic moment of a level of a term depends on S, 
L, and J.) Construct a diagram showing the expected 
anomalous Zeeman effect for the transition 3 D — ► 3 F. Mark 
the polarization of the lines. What will happen to the spectrum 
when the field is markedly increased (to about 100 kG>7 

Further information. See MQM Chapter 8 for a more detailed 
discussion. Accounts of the Zeeman effect, and the use to 
which it can be put, will be found in §11.3 of Herzberg 
(1944), §3.15 of King (1964), §ll I. A3 and gill. F of Kuhn 
(1962), and Chapter XVI of Condon and Shortley (1963). 



265 



TABLE 1 

Physical properties of benzene 



C— C bond length 
C— H bond length 
enthalpy of 
formation 
resonance energy 

first ionization 
potential 

polarizability 



refractive index 
(20°C, D-line) 

relative permittivity 
(20°C) 

magnetic 
susceptibility 



AW f ° 



a. 



139-7 pm (1-397 A) 
108-4 pm( 1-084 A) 

83-2 kJ mof 1 (19-820 kcal mol" 1 ) 
150 kJ mof 1 (36-0 kcal mol" 1 ) 

9-24 eV 

= 6-35X 10" M cm 3 

(6-67 X 10 _4I Fm _2 ) 

= 12-31 X 10 _24 cm 3 

(10-89 X 10 _4l Fm" 2 ) 

= 10-32X 10 _24 cm 3 

(9-13 X lO^Fm -2 ) 



XI. 

x± 



1-5011 

= 2-284 

= -3-49X 10~ s 
= -9-46X 10~ 5 
= -7-47X 10~ s 



absorption bands 6-8eV(E 



lu 

6-0eV(B lu ?< 



A lg );6-0eV(E 2g - 
-A lg );4-9eV(B^ 



■v 

" A lg ) 



TABLE 2 

Bond-order— bond-length correlations 



Hydrocarbon 


Bond 


w-bond order 


Bond distance (pm) 


ethene 




10 


133-5 


benzene 




0-667 


139-7 


naphthalene 


1,2 


0-725 


136-5 




2,3 


0-603 


140-4 




1,9 


0-554 


142-5 




9,10 


0-518 


139-3 


anthracene 


1,2 


0-738 


1370 




2,3 


0-586 


140-8 




1,9a 


0-535 


142-3 




9,9a 


0-606 


139-6 




4a,9a 


0-485 


143-6 


graphite 




0-535 


142-1 



See Streitweiser (1961) and Daudel, Lefebvre, and Moser 
(1959) for more information and analysis. 



266 



'5v 



2C 5 



TABLE 3 

Character tables 



C2v 

(2mm) 


E 


c 2 


o (xz) 


o'M) 




Ai 


1 


1 


1 


1 


z 


x 2 ,y 2 ,z 2 


k 


A 2 


1 


1 


-1 


-1 


R : 


xy 




Bi 


1 


-1 


1 


-1 


X.Ry 


xz 




B 2 


1 


-1 


-1 


1 


y- R x 


yz 






(4mm) 


E 


2C 4 


c 2 


2a 

V 


20 d 




Ai 


1 


1 


1 


1 


1 


z 


x 2 + y\z 2 


A 2 

B! 


1 
1 


1 
-1 


1 
1 


-1 
1 


-1 
-1 


R z 


x 2 -y 2 


B 2 


1 


-1 


1 


-1 


1 




xy 


E 


2 





-2 








(x,y)(R x ,R y ) 


(xz, yz) 



2Ci 



5a 



A. 
A 2 
Ei 
E 2 



1 

1 
2cos72° 
2cos144° 



1 

1 
2cos144° 
2cos72° 



1 

-1 







z 
{x,y)(R x .R y ) 



x 2 + y 2 ,z 2 

(xz, yz) 
(x 2 -y 2 .xy) 







2cos72 


= 0-61803 


2 cos 


144 




-1-61803 








(6mm) 


E 


2C 6 


2C 3 






c 2 


3a 

V 


3 °d 




A, 


1 










1 


1 


1 


z 


x 2 + y 2 ,z 2 


A 2 


1 










1 


-1 


-1 


R z 




B, 


1 


—1 








-1 


1 


-1 






B 2 


1 


—1 








-1 


-1 


1 






E, 
E 2 


2 
2 


—1 


—1 






-2 
2 










(x,yHR x ,R y ) 


(xz, yz) 
(x 2 -y 2 ,xy) 



267 



c, v 


E 


2Cf 


°°a 

V 




A„ir 

A 2 ,2T 

E,.n 

E 2 ,A 
E 3 ,$ 


1 
1 
2 
2 
2 


1 

1 
2cos0 
2 cos 20 
2 cos 30 


1 
-1 





z 

R z 

te,y)(R x ,R Y ) 


x 2 + y 2 ,z 2 

(xz, yz) 
{x 2 — y 2 ,xy) 




e = exp (27TJ/3) 



(x,y,z)(R x ,R v ,R z ) 



x 2 + y 2 + z 2 
(x 2 -y 2 ,2z 2 -x- 
[xy, xz, yz) 



T d 
(4 3m) 


E 


8C 3 


3C 2 


6S 4 


6 °d 




A, 


1 


1 


1 


1 


1 




x 2 + y 2 + z 2 


A 2 


1 


1 


1 


-1 


-1 






E 


2 


-1 


2 










C2z 2 -x 2 -y 2 ,x 2 -y 2 ) 


Ti 
T 2 


3 
3 






-1 
-1 


1 


-1 
1 




(xy, xz, yz) 




















(432) 


E 


8C 3 


3C 2 


6C 4 


6C 2 






Ai 


1 


1 


1 


1 


1 




x 2 + y 2 + z 2 


A 2 


1 


1 


1 


-1 


-1 






E 


2 


-1 


2 










(2z 2 -x 2 -y 2 ,x 2 - y 2 ) 


T! 


3 





-1 


1 


-1 


ix,y,z)(R x .R y ,R z ) 




T 2 


3 





-1 


-1 


1 




(xy,xz,yz) 



268 



TABLE 4 

T-values for selected molecules 



CH 3 C0 2 H* 


-1-37 


CH3I 


7-84 


CH 3 CH*0 


0-20 


CH3COCH3 


7-91 


C6H 6 


2-73 


CH3C0 2 H 


7-93 


p-C 6 H|(CH 3 ) 2 


305 


CH 3 CN 


803 


CgHs 


4-31 


C 2 H 2 


8-51 


C2H4 


4-68 


CeHi 2 


8-56 


H 2 (0°C) 


4-68 


C 2 H6 


9-11 


H 2 


5-66 


H 2 0(vapour) 


9-26 


C 6 H 5 0CHJ 


6-27 


CH 4 


9-86 


CHJOH 


6-62 


Si(CH 4 ) 4 


1000 (definition 


CgHsCHjCHs 


7-38 


HCI 


10-31 


C6H5CH3 


7-66 


HBr 


14-21 


CHJCHO 


7-80 


HI 


23-11 



TABLE 5 

Colour, frequency, and energy of lights 



Colour 


Wavelength 
(nm) 


Frequency 
(Hz) 


Wave number 
(cm' 1 ) 




Energy 






" 














(eV) 


(kJ mol" 1 ) 


(kcal mol -1 ) 


infrared 


1000 


300 X 10 14 


1-00X 10 4 


1-23 


120 


28-5 


red 


700 


4-28 


1-43 


1-77 


171 


40-8 


orange 


620 


4-84 


1-61 


200 


193 


46-1 


yellow 


580 


5-17 


1-72 


2-14 


206 


49-3 


green 


530 


5-66 


1-89 


2-34 


226 


53-9 


blue 


470 


6-38 


2-13 


2-63 


254 


60-8 


violet 


420 


7-14 


2-38 


2-95 


285 


68-1 


near ultraviolet 


300 


100X 10 1S 


3-33 


4-15 


400 


95-7 


far ultraviolet 


200 


1-50 


500 


6-22 


600 


143 



t Adapted from Calvert and Pitts (1966). 



269 







TABLE 6 








Dipole moments (debyes) 






NH 3 


1-47 


CHCI 3 


101 


H 


H 2 


1-85 


CH 2 CI 2 


1-57 


21 


HF 


1-91 


CH3CI 


1-87 


Li 


HCI 


108 


CH3OH 


1-71 


10 


HBr 


0-80 


CH3CHO 


2-72 


Na 


HI 


0-42 


CH3OCH3 


1-30 


0-9 


HCN 


30 


CH 3 CH 2 OH 


1-68 


K 


N0 2 


0-32 


C6H5CH3 


0-36 


o- 


N 2 


0-17 


o-C6H4(CH 3 ) 2 


0-62 




S0 2 


1-59 


C 6 H 5 CI 


1-57 




CO 


0-10 


C 6 H 5 Br 

O-C6H4CI2 


1-70 
2-25 





/77-C 6 H 4 CI 2 1-72 

TABLE 7 

Pauling electronegativities 



H 














2-1 














Li 


Be 


B 


C 


N 





F 


10 


1-5 


20 


2-6 


30 


3-4 


40 


Na 


Mg 


Al 


Si 


P 


S 


CI 


0-9 


1-3 


1-6 


1-9 


2-2 


2-6 


3-2 


K 






Ge 


As 


Se 


Br 


0-8 






20 


2-2 


2-6 


30 


Rb 












I 


0-8 












2-7 


Cs 














0-8 















2 



Dipole moment (in debyes) ju AB ~X A — Xq 
Ionic character (per cent) 16ix A ~Xb ' + 3 ' 5l X A ~ Xb ' 
Covalent-ionic resonance energy (in eV) A~ (X A ~X e ) 
Mul liken scale M^ — M B = 2-78 (X A ~" Xg) 

For a complete list of Pauling and Mulliken electronegativities 
see p. 1 14 of Cotton and Wilkinson (1972). 



270 



TABLE 8 

Oscillator strengths and molar extinction coefficients 







f 


e/cm -1 dm 3 mol 


electric dipole allowed 


1 


10 4 - 10 5 


magnetic dipole allowed 


10~ 5 


10" 2 - 10 


electric quadrupole allowed 


10" 7 


10" 4 - 10" 1 


spin forbidden (S-T) 


10" 5 


10" 2 - 10 


parity forbidden 


10" 1 


10 3 


Examples 












W nm 


e/cm -1 dm 3 mol 


ir**-ir 


C=C 


180 


10 000 




c=c-c=c 


220 


20 000 


7T*^n 


oo 


280 


20 




N=N 


350 


100 




c=c-c=o 


320 


100 




N=0 


660 


10 



271 



Isotope 

»H 
2 H 

6 Li 
7 Li 
9 Be 



"r 



31 D 



33o 



35, 



37, 



41 



K 



79, 



81 



Abundance 
(per cent) 



99-9844 
00156 

7-43 

92-57 

100 

18-83 

81-17 

1-108 

99-635 

0-365 

0037 

100 

100 
0-74 
75-4 
24-6 

9308 

6-91 

50-57 

49-43 



TABLE 9 

Hyperfine fields and spin-orbit coupling in some atoms 

|2 



Spin 

2 
1 



2 
3 

2 
3 
2 
3 
2 

3 

2 
3 

2 
3 

2 
3 

2 



(a.u.) 



0-1673 



2 


0-5704 


3 


1-408 


3 

2 




1 

2 


2-767 


1 


4-770 


1 

2 




5 
2 


7-638 


1 
2 


11-966 



5-6251 

7-9187 

10-6435 



"P 
(a.u.) 



0-775 

1-692 
3-101 

4-974 
7-546 

3-3187 
4-8140 
6-7095 



19-4127 11-8758 7800 456 

564 
This Table is adapted from Atkins and Symons (1967) and Morton, Rowlands, and Whiffen (1962). 



Isotropic 

coupling 

<G) 

508 
78 

39 
105 
130 
242 
725 

1130 
552 
775 

1660 
17 200 

3640 

975 

1680 

1395 

83 
45 
7800 
8400 



Anisotropic 

coupling 

(G) 



38 
66 
34 
48 
104 
1084 

206 
56 

100 
84 



i 

(cm-i) 



0-2 

1 
11 

29 
76 

151 
270 

299 
382 
586 

38 
2460 



272 









TABLE 10 










Diatomic molecules 






Molecule 


Vibration 

frequency 

(cm" 1 ) 


Bond length 
(pm) 


Rotation 

constant 

(cm-i) 


Dissociation 

energy 

(eV) 


Force-constant 
(N m _1 ) 


Br 2 ( 79 Br 8I Br) 


323-2 


228-3 


008091 


1-971 


245-8 


% 


1641-35 


131-17 


1-6326 


3-6 


952-1 


3S CI 2 


564-9 


198-8 


0-2438 


2-475 


328-6 


12 C 16Q 


2170-21 


112-81 


1-9313 


11-108(?) 


1902 


19 F 2 


892-1 


143-5 




<2-75 


718-6 


'Ha 


4395-2 


74-16 


60-809 


4-476 


573-4 


2 H 2 


3118-4 


74-16 


30-429 


4-553 


576-8 


'HBr 


2649-67 


141-3 


8-473 


3-75 


411-6 


1 H 3S C , 


2989-74 


127-460 


10-5909 


4-430 


515-7 


l H !9p 


4138-52 


91-71 


20-939 


<6-40 


965-5 


1^27, 


2309-5 


160-4 


6-551 


3056 


314-1 


127i 


214-57 


266-6 


003735 


1-5417 


172-1 


14 N 2 


2359-61 


109-4 


2010 


9-756 


2296 


16 2 


1580-361 


120-739 


1-44566 


5080 


1177 



These molecules have been selected from a longer list compiled by Herzberg (1950). 



273 



TABLE 11 

Hermite polynomials and oscillator wavefunctions 

Harmonic oscillator: mass m, force-constant k 
Schrodinger equation 

-(h 2 /2m)d 2 0(x)/dx 2 + \kx 2 4>W = £^(x); 
energies 

i>= 0,1,2,...; co =(Ar//n) ,/l ; 
wavefunctions 

^ )= (i) M ^ ,W/21 
a = mcj fh, f = / (mCob/h)x = a Vl x, W,(f) are Hermite 

polynomials. 
Properties of Hermite polynomials 
H v ($) satisfies 

Hl($)-2$H' v ($) + 2vH v ($) = 0. 
Rodrigues' formula 

W„(f) = (-1)"exp? 2 £ exp(-f 2 ). 

dr 



Recursion relations 

K® = 2VH v-X® 

Integrals 

Cdrexp(-r 2 )^(f) Wl) ,(r) = |^^^ 



Explicit forms 






"o(f ) = 


= 1 






"i(?> = 


= 2f 






"atf) = 


= 4f 2 - 


2 




" 3 <f) = 


= 8f 3 - 


12? 




«4(J) = 


= 16J*- 


-48f 2 + 12 




" 5 (?) = 


= 32f 5 - 


-160f 3 +120f 




w 6 <r> = 


= 64^" 


-480^ + 720^-120 




«7<J1 = 


= 128f 7 


-1344f s + 3360f 3 - 


1680f 


««tf) = 


= 256? 8 


-3584$* + 13440? 4 - 


-13440f 2 + 1680 







TABLE 12 










Some Debye temperatures of solids 






d D /K 




d D /K 




e D /K 


Li 


344 


Cu 


343 


Al 


428 


Na 


156 


Ag 


226 


Ti 


420 


K 


91-1 


Au 


162 


Hg 


71-9 


Rb 


55-5 










Cs 


39-5 










C (diamond) 


2230 










NaCI 


321 










KCI 


230 










KBr 


177 










CaF 2 


570 











274 



Coordination 
number 

2 

3 



TABLE 13 

Hybrid orbitals 

Shape 

linear 
bent 

trigonal planar 
unsymmetrical planar 
trigonal pyramidal 
tetrahedral 
irregular tetrahedral 
tetragonal pyramidal 
bi pyramidal 
tetragonal pyramidal 
pentagonal planar 
pentagonal pyramidal 
octahedral 
trigonal prismatic 
trigonal antiprismatic 



Hybridization 

sp, dp 

p 2 , ds, d 2 

sp 2 , dp 2 , ds 2 , d 3 

dsp 

P 3 .d 2 P 

sp 3 , d 3 s 

d 2 sp, dp 3 , d 3 p 

d 4 

dsp 3 , d 3 sp 

d 2 sp 2 , d 4 s, d 2 p 3 , d 4 p 

d 3 p 2 

d s 

d 2 sp 3 
d 4 sp, d 5 p 
d 3 p 3 



TABLE 14 

Associated Laguerre polynomials and functions 

The Laguerre differential equation is 

x(d 2 f/dx 2 ) + (1 -x){df/dx) +nf= 0, 
with n a non-negative integer. Its solutions are the Laguerre 
polynomials L n (x), 

^=M,»!(tfi')(^)v- 

= e*(d/dx)"x'V x . 

The associated Laguerre polynomials are related to L n (x) by 
differentiation: 

and satisfy 

x(d 2 f/dx 2 ) + (k + 1 -x)(df/dx) + in - k)f = 
The associated Laguerre functions are related to L k (x) as 



L k U) = d k L n (x)/dx k 



J*{x) = e- y ' x x ,Mk - 1) L k „{x) 



follows: 

- n r~, - -„. 

and satisfy the equation 

xldV/dx 2 ) + 2(df/dx) + [n-±(k- 1) - \x - (k 2 - 1)/4x]^=0. 
The normalized radial components of the hydrogen-atom wave 
functions are 



with p = 2Zr/na Q 

These wavef unctions are developed in Table 15. 






(P), 



275 



TABLE 15 

Hydrogen -atom wavefunctions 
General form: 

R n(r) are proportional to the associated Laguerre functions 
(see Table 14) and the Yn (0, <j>) are the spherical harmonics 
(see Table 23). 

Specific form of radial equation; p = 2Zr/na . 

1s: R 10 (r) = IZ/aoJ^af" 3 

2s: R 20 {r) = (Z/a ) 3/2 (1/2/2)(2-p)e- p/2 

2p: /? 21 (r) = (Z/a ) 3/2 (1/2/6)pe-" /2 

3s: /?3o(/-) = (Z/a ) 3/2 (1/9/3)(6 -6p + p 2 )e" p/2 

3p: /?3i(r) = (Z/a ) 3/2 (1/9/61(4 -pipe"" 72 

3d: R 32 (r) = (Z/a ) 3/2 (1/9/30)pV p/2 

4s: /? 40 (r) = (Z/a ) 3/2 (1/96)(24- 36p+ 12p 2 -p 3 )e"" p/2 

4p: /? 41 (r) = (Z/a ) V2 ( 1/32/ 15) (20-1 Op + p 2 )pe _p/2 

4d: /? 42 (r) = (Z/a ) 3/2 ( 1/96/ 5) (6 -p)pV> /2 

4f: R 43 (r) = (Z/a ) 3/2 (1/96/35) pV p/2 



276 



TABLE 16 

The hydrogen atom 

Experimental data 
Spectral lines (Vnm): 

Lyman series: 121567(a), 102-572(/3), 97-253(7) 91-15 

Balmer series: 65628(a), 486-13(j8), 434-05(7) 364-6 

Paschen series: 1875-1, 1281-8, 1093-8, . . ., 820-4 

Brackett series: 4051-2, 2625-1 1458-4 

Pfund series: 7451-2 . . ., 2278-8 
Humphreys series: 12 368, . . ., 3281-4. 

Ionization potential: 1097 X 10 s cm -1 , 13-60 eV. 

Electron affinity: 0-77 eV. 

Lamb shift: (2S V -2P„): 1-058 GHz. 

At Vi 

Hyperfine interaction (Fermi contact interaction): 
1420-4 MHz. 

Polarizability (ground state): 4-5 a%, 8-7 X 10~ 31 m 3 . 

Diamagnetic susceptibility (ground state): —3-97 X 10" 6 . 

Covalent radius: 30 pm (0-30 A). 

Electronegativity (Pauling): 2-1. 

Theoretical data 

Hamiltonian for the atom: H = — (h 2 /2p)V 2 — e 2 /47re r. 

Energy of state with quantum numbers n, 2, m»: 

E o _ / Pe< \ 1 _ An 

n= 1,2, 3, ...;£ = 0, 1,2 /7-1;/t? 6 = £, £-1 -£. 

p is the reduced mass m m Hm + m ) and /?„ is the 

•^ e p e p H 

"Rydberg constant. 

Degeneracy of state with energy — R In 2 : n 2 . 

Wavefunctions ty nim Ar'. 9. </>) = R n %(r) Y^ (0, 0). 



The angular functions are the "spherical harmonics, see 
Table 23; the radial functions are the associated Laguerre 
functions 



R ni 1r) «e-'V A 2 « +1 



(P) 



where p = 2Zr/na and L (p) is an associated Laguerre 
polynomial. See Tables 14 and 15 for their analytical form 
and normalization. 



Expectation values, etc. 
Mean radius, etc: t 
(r 2 )=(aW/Z 2 )h + 



3 (, 3£(£+1)-A 

l(1-^i)j 



<r)={aon 2 /Z) J1+-, 

<r- 1 ) = Z/a n 2 

<r- 2 > = Z 2 /agA7 3 (£ + l) 

<r- 3 >=(Z/a ) 3 /n 3 £(£ + |)(£+1). 

Most probable radius (ground state): a (Bohr radius, 
5-291 771 5X 10" n m). 

Spin-orbit coupling parameter: 




\87re /7? 2 c 2 ao 

u 2 z*Rjr 2 

/7 3 £(£+j)(£+1) ' 

a is the "fine-structure constant. 
Probability at the nucleus: \\jj (0)l 2 : 



/? 3 £(£ + 5-)(£+ 1). 



■Z 3 /ira 3 n 3 . 



277 



Isotope 
(*: radioactive) 



l H 



3 H* 



y Be 



io D 



14, 



N 



N 



is, 

19 F 



23, 



Na* 
Na 



3S S* 



35 CI 
36 C| . 

37 CI 

39 K 

40 K* 



43, 



79, 



'Ca 

Br 

*Br 







TABLE 17 




Selected nuclear-spin properties 


Natural 
abundance 
(per cent) 


Spin 


Magnetic 
moment 


- 


1 

2 


-1-9130 


999844 


1 

2 


2-79270 


1-56 X 10" 2 


1 


0-85738 


- 


1 

2 


2-9788 


7-43 


1 


0-82191 


92-57 


3 
2 


3-2560 


100 


3 

2 


-1-1774 


18-83 


3 


1-8006 


81-17 


3 

2 


2-6880 


1-108 


1 
2 


0-70216 


99-635 


1 


0-40357 


0-365 


1 
2 


-0-28304 


3-7 X 10~ 2 


5 

2 


-1-8930 


100 


1 

2 


2-6273 


- 


3 


1-745 


100 


3 

2 


2-2161 


100 


1 

2 


1-1305 


0-74 


3 
2 


0-64274 


- 


3 

2 


100 


75-4 


3 
2 


0-82089 


- 


2 


1-2838 


24-6 


3 

2 


0-68329 


9308 


3 

2 


0-39094 


1-19 X 10~ 2 


4 


-1-294 


6-91 


3 

2 


0-21453 


0-13 


7 
2 


-1-3153 


50-57 


3 

2 


20990 


49-43 


3 

2 


2-2626 



Electric 

quadrupole 

moment 

(e X 10-24cm~2) 



2-77 X 10~ 3 

4-6 X 10" 4 
-4-2 X 10~ 2 
2X 10" 2 

0-111 
3-55 X 10~ 2 

2X 10~ 2 

-4X 10" 3 



0-1 

-6-4 X 10~ 2 

4-5 X 10~ 2 

-7-97 X 10" 2 

-1-68X 10" 2 

-6-21 X 10" 2 

? 

? 

? 

? 

0-33 

0-28 . 



n.m.r. frequency 

at 10 kG 

(MHz) 

29-165 
42-577 

6-536 
45-414 

6-265 
16-547 

5-983 

4-575 

13-660 

10-705 

3076 

4-315 

5-772 

40055 

4-434 

11-262 

17-235 

3-266 

508 

4-172 

4-893 

3-472 

1-987 

2-470 

1092 

2-865 

10-667 

11-498 



278 



TABLE 18 

First and second ionization potentials (in eV) of some elements 



H 
13-599 














He 
24-588 
54-418 


Li 
5-392 
75-641 


Be 

9-323 
18-211 


B 

8-298 

25-156 


C 
11-260 
24-383 


N 
14-53 
29-602 



13-618 
35-118 


F 
17-423 
34-98 


Ne 
21-565 
40-964 


Na 

5-139 
47-29 


Mg 

7-646 
15035 


Al 

5-986 
18-828 


Si 
8-152 
16-346 


P 
10-487 
19-72 


S 
10-360 
234 


CI 

12-967 
23-80 


Ar 
15-760 
27-62 


K 
4-341 
31-81 


Ca 

6-113 
11-872 










Br 
11-814 
21-6 


Kr 
14000 
24-56 



279 



Gas lasers 
He-Ne 



Ne 
Ar 



Kr 



Xe 

H 2 
N 2 
C0 2 

Solid-state lasers 
Ruby 



Nd 3+ :YAG 



GaAs 
In As 



TABLE 19 






Some laser systems 






\/nm 


Power 


Mode and duration of pulse 


632-8 


100 mW 


CW 


1084 


10 mW 


CW 


1152 


20 mW 


CW 


3391 


10 mW 


CW 


540-1 


10 kW 


pulsed (3 ns) 


4880 


10W 


CW 


514-5 


500 mW 


CW 


457-9 


40 mW 


CW 


465-8 


- 


CW 


472-7 


— 


CW 


476-5 


100 mW 


CW 


496-5 


100 mW 


CW 


647-1 


5W 


CW 


568-2 


50 mW 


CW 


520-8 


50 mW 


CW 


476-2 


25 mW 


CW 


1700 


— 




1160 


— 




337-1 


0-2 MW 


pulsed (10 ns) 


10600 (10-6 Mm) 


1 kW 


CW 


694-3 


400 kW 


conventional pulsed (1 ms 




16 GW 


Q-switched (10-20 ns) 




16 GW 


mode-locked (10 ps) 


10600 


300 W 


CW 




10 MW 


Q-switched (10 ns) 




fOMW 


mode-locked (1 ps) 


9050 


60 W 


pulsed (200 ns) 


8450 


1 W 


pulsed (2 jus) 


3150 


50 mW 


pulsed (2 ns) 



280 



TABLE 20 

Maxwell equations 

Basic definitions 

E : electric-field intensity (V m _1 ) 

H : magnetic-field intensity (Am -1 ) 

D : electric displacement (C m~ 2 ) 

B : magnetic induction, flux density (T or Wb rrf 2 ) 

p : charge density (C m" 3 ) 

J : current density (A m -2 ). 

D and B may be related to E and H respectively, through the 
polarization P and magnetization M: 

D = e E + P B = jUoH + /UoM (/ioe = c -2 ). 
Maxwell equations 

V.D =p 
V.B =0 

VAE = -bB/bt 
VAH = J + dD/dt. 

Potentials 

A : vector potential, : scalar potential 
B = VAA 

E = -dA/df - V0. 

Gauge transformation : if A — ■> A + Vx and — *■ — dx/3 i. 
where x is any differentiable function, E and B are unchanged. 
When x is chosen so that V.A = we are in the Coulomb gauge 
and if V.A + (1/c 2 )(90/3r) = we are in the Lorentz gauge. 

TABLE 21 

Slater atomic orbitals 



Values of Z_, x 

eff 


= Z—O for s. 


p-orbitals of the neutral first- 


and second-row atoms. 








1s 


H 
1 














He 
1-70 




Li 


Be 


B 


C 


N 





F 


Ne 


1s 


2-70 


3-70 


4-70 


5-70 


6-70 


7-70 


8-70 


9-70 


2s,2p 


1-30 


1-95 


2-60 


3-25 


3-90 


4-55 


5-20 


5-85 




Na 


Mg 


Al 


Si 


P 


S 


CI 


Ar 


Is 


10-70 


11-70 


12-70 


13-70 


14-70 


15-70 


16-70 


17-70 


2s, 2p 


6-85 


7-85 


8-85 


9-85 


10-85 


11-85 


12-85 


13-85 


3s, 3p 


2-20 


2-85 


3-50 


4-15 


4-80 


5-45 


6-10 


6-75 



281 



TABLE 22 

Spherical harmonics and Legendre functions 

The spherical harmonics VV<0, 0) satisfy 

A 2 ^ =-«<£ + D/^ ^VA'.'.'.-e. 

Factorization 

/^ are the associated Legendre functions: 



^(x) = [(1-x 2 ) m/2 /2 £ C! 






<x 2 -1)* 



(1 -x 2 )" 



dx" 1 



*>(*> 



1) c . 



Pj are the Legendre functions: 

/' B (x) = (1/2 B £l)f- i i-JUf 2 - 

Properties of Vn 
symmetry: 

y^ (7T- 0, + ir) = (-1 ) £ v^ (0, 0) 



orthonormality: 

;; d 0sin0 £* d^ v. w^id, <t» = o^s^- 

(5no'5 ' = 1 if both 2 = £' and m = m ', and otherwise), 
xx mm 

Recursion relations for Pg": 

(£-/n + D/^ +1 <x) - (2B + Dx/^W + (« + mV»g! I U) = 

x/f(x) - (£ -/n + Dd -x 2 )*Af _1 (x) -/»£, (x) = 

/f tl M -xP^U) - (C + m)d -x 2 ) 14 ^" 1 (x) = 

(£-/n + D/^ +1 W + (1 -x 2 )*Ff +l U) - 
-{C + /r7 + 1)x/' c T '(x) = 

(1 -x 2 ) 34 /^* 1 (x) - Zrox/'g 1 (x) + 

+ (x + m)(x-m+1)(1-x 2 ) , V(r l (x) = 

(1 -x 2 )* m/fM = (C + Dx/fU) - (fi-m + DPg^lx) 

= (fi + m)Pg: i (x)-ex/ , K "{x). 
Integrals of P™\ 

f^dxfffWPftb) = 26 M »[(£ + m)\IW + 1)(« -/n)l] 



282 



£ 

1 



m 



±1 



±1 

±2 


±1 
±2 
±3 



TABLE 23 

Cartesian and polar forms of spherical harmonics VV, 

Cartesian form 
1/27I 54 

jO/n)* <*//■) 
4(3/27T) ,/ '(x±iK)/r 

i(5/7r)' /2 (2z 2 -x 2 -/V 2 
+~mi2n) y *z(x±\y)lr 2 



£(15/27r) ya (x±iy) 2 /r 2 

5-(7/7r) ya z(2? 2 -3x 2 -3y 2 )//- 3 
+|(21/7r) ,/a (x ± \y)(4z 2 -x 2 -y 2 )/r 3 
£(105/27T) ,/a z(x iiy) 2 //- 3 
4(35/7r) ,/a (x+i/) 3 //- 3 



Polar form 



1/27I 34 
5-<3/7T) y2 cos0 



-1 



_(3/27r)' /a sin0e ±i0 
j(5/7r) ,/j (3cos 2 0-1) 
+j(15/27r) ,/a cos0sin0e ±i0 
£<15/27r) ya sin 2 0e ±2i * 
j(7/7r)' /a (2cos 3 - 3cos0 sin 2 0) 
4(21/7T)' /a (4cos 2 sin0 - sin 3 0)e ±i0 
£(1 05/27^0)50 sin 2 0e ±2i0 
4(35/ir) ,/a sin 3 0e ±3i * 



283 





TABLE 24 






Vibration frequencies 




CH stretch 






2850-2960 cm" 1 


CHbend 






1340-1465 


C— C bend and stretch 




700-1250 


C=C stretch 






1620-1680 


C^C stretch 






2100-2260 


0— H stretch 






3590-3650 


H— bonds 






3200-3570 


C=0 stretch 






1640-1780 


N— H stretch 






3200-3500 


C^N stretch 






2215-2275 


C=N stretch 






1480-1690 


N=N stretch 






1575-1630 


P— H stretch 






2350-2440 


C-F stretch 






1000-1400 


C-CI stretch 






600-800 


C— Br stretch 






500-600 


C— 1 stretch 






~500 


cor 






1410-1450 


N0~ 3 






1350-1420 


NO2 






1230-1250 


sor 






1080-1130 


silicates 






900-1100 


See Bellamy (1958, 1968) for < 


i thorough analysis. 



TABLE 25 

Photoelectric (and thermionic) work functions (eV) 



Li 




Be 




2-42 




3-9 
(367) 




Na 




Mg 




2-3 




3-7 




K 


Cu 


Ca 


Zn 


2-25 


4-8 


3-2 


4-3 




(4-5) 


(2-2) 




Rb 


Ag 




Cd 


209 


4-3 
(43) 




4-1 


Cs 


Au 






2-14 


5-4 






(1-86) 


(43) 







284 





TABLE 26 






TABLE 27 






Covalent radii 




Vector properties 


Element 




fl/pm 


We deal with the vectors F and G, where 


H 

c- 

C= 




30 
77 
67 


F = F \ + F \ + F k 

x y' z 

G = G x i + G / j + G z k. 


C= 




60 


Scalar product 


C— (benzene) 

o- 

0= 




69-5 

66 

57 


F.G = FG +F G +F G . 
xx y y z z 

Vector product 


N— (amino) 
N— (nitrate) 
N= (nitrate) 




70 
70 • 
65 


FAG = 


i j k 

F F F 
x y z 

G G G 
x y z 


= -GA F. 


s- 

S= (sulphate) 

ci— 




104 
95 
99 


Triple and quadruple products 

F.(G A H) = G.(H A F) = H.(F A G) = (F A G).H 


Br- 




114 


F A (G A H) = G(F.H) - H(F.G) 


1- 




133 


(F A G) . (H A 1) = (F.H)(G.I) - (F.I)(G.H) . 

divergence 

divF = V.F=(^j + ^ + ^, a scalar. 

curl 

1 i J k 1 
curl F = V A F = 1 3/3* 3/3/ 3/3* | , a vector. 










\ F x F y 


F z \ 



differentiation of products: 
V.(UF) = UV.F + F.Vt/ 

V A (i/F) = L/(V A F) + (VLO A F 
V.(F A G) = G.(V A F) - F.(V A G) 
\7A(VAF) =V(V.F)-\7 2 F 

V A (F A G) = F(V.G) - (V.F)G + (G.V)F - (F.V)G 
V(F.G) = (F.V)G + (G.V)F + F A (V A G) + G A (V A F) 
V 2 U= V.VU 

V A (Vi/) = . 



Bibliography 



MQM refers to 

ATKINS, P.W. (1970). Molecular quantum mechanics. Clarendon Press, 

Oxford. 
Volumes in the Oxford Chemistry Series (denoted OCS) which are now 
available and which elaborate many of the points in the text, are as 
follows: 

1. McLAUCHLAN, K.A. (1972). Magnetic resonance. 

2. ROBBINS, J. (1972). Ions in solution (2): an introduction to 
electrochemistry. 

3. PUDDEPHATT, R.J. (1972). The periodic table of the elements. 

4. JACKSON, R.A. (1972). Mechanism: an introduction to the study 
of organic reactions. 

5. WHITTAKER, D. (1973). Stereochemistry and mechanism. 

6. HUGHES, G. (1973). Radiation chemistry. 

7. PASS, G. (1973). Ions in solution (3): inorganic properties. 

8. SMITH, E.B. (1973). Basic chemical thermodynamics. 

9. COULSON, C.A. (1973). 77>e shape and structure of molecules. 

10. WORMALD, J. (1973). Diffraction methods. 

11. SHORTER, J. (1973). Correlation analysis in organic chemistry: 
an introduction to linear free-energy relationships. 

12. STERN, E.S. (Ed.) (1973). The chemist in industry (1): fine 
chemicals for polymers. 

13. EARNSHAW, A. and HARRINGTON, T.J. (1973). The chemistry 
of the transition elements. 

14. ALBERY, W.J. (1974). Electrode kinetics. 

16. FYFE, W.S. (1974). Geochemistry. 

17. STERN, E.S. (Ed.) (1974). The chemist in industry (2): human 
health and plant protection. 

18. BONO, G.C. (1974). Heterogeneous catalysis: principles and 
applications. 

19. GASSER, R.P.H. and RICHARDS, W.G. (1974). Entropy and 
energy levels. 

20. SPEDDING, D.J. (1974). Air pollution. 

21. ATKINS, P.W. (1974). Quanta: a handbook of concepts. 

22. PILLING, M.J. (1974). Reaction kinetics. 

Volumes in the Oxford Physics Series (OPS) now available are 

1. ROBINSON, F.N.H. (1973). Electromagnetism. 

2. LANCASTER, G. (1973). D.c. and a.c. circuits. 

3. INGRAM, D.J.E. (1973). Radiation and quantum physics. 

5. JENNINGS, B.R. and MORRIS, V.J. (1974). Atoms in contact 

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